It depends. First I'll tackle the myth, which is pretty persistent, namely that Reagan's military spending (or sometimes strictly that related to the Strategic Defense Initiative aka "Star Wars") bankrupted the USSR and ended the Cold War. A repost of an old answer of mine:

Part I

The short answer is that while the Soviet Union did collapse in no small part because of budget deficits and economic stability, and while SDI did play a complicated role in arms control negotiations towards the end of the Cold War, responses to SDI were not a major factor in either the collapse of the Soviet Union, nor in the end of the Cold War.

First, about the Strategic Defense Initiative. SDI, simply, was a defense program that was supposed to render nuclear weapons obsolete by creating a system of anti-ballistic missiles (or lasers) that would be able to intercept any Intercontinental Ballistic Missiles with nuclear warheads fired at the United States or its allies. The first call for such a program was in President Reagan’s “Address to the Nation on Defense and National Security”, given on March 23, 1983:

”What if free people could live secure in the knowledge that their security did not rest upon the threat of instant U.S. retaliation to deter a Soviet attack, that we could intercept and destroy strategic ballistic missiles before they reached our own soil or that of our allies? I know this is a formidable, technical task, one that may not be accomplished before the end of this century. Yet, current technology has attained a level of sophistication where it's reasonable for us to begin this effort. It will take years, probably decades of effort on many fronts. There will be failures and setbacks, just as there will be successes and breakthroughs. And as we proceed, we must remain constant in preserving the nuclear deterrent and maintaining a solid capability for flexible response. But isn't it worth every investment necessary to free the world from the threat of nuclear war? We know it is.”

Now, while this was a momentous announcement, it is largely a concluding section to a larger speech, one that effectively is given to justify increased US military spending since Reagan came to office in 1981. The general thrust of the speech was: “the Soviets have increased their military spending and research since the 1970s, the US has fallen behind, and needs to spend more to catch up.” Small note: while it has been argued, with some documentary evidence from Reagan’s diary, that the film “The Day After” had a profound influence on his desire to eliminate the nuclear threat, that made-for-TV film was broadcast in November 1983, some eight months after this national address.

Congress appropriated $1.39 billion for the initiative in 1984, but this was largely for research. The project was considered to have a final cost of $70 billion, soon rising to $170 billion, with no operational defense before 2000. Ultimately SDI was renamed in 1993, and then reorganized again in 2002 as the Missile Defense Agency. While it continues to conduct anti-ballistic missile research, the results have been mixed, and to date there is no ballistic missile shield rendering nuclear weapons obsolete.

So, so much for SDI. Now let’s look at the Soviet response to the program. The impact that the announcement of SDI had on Soviet strategic thinking has been debated. First, it’s worth noting that the Soviet defense industry and the Politburo did plan responses to SDI:

A decision of the Central Committee and the Council of Ministers of 15 July 1985 approved a number of "long-term research and development programs aimed at exploring the ways to create a multi-layered defense system with ground-based and space-based elements." It should be noted that no commitment to deployment of any of these systems was made at the time. The goal of the research and development effort was "to create by 1995 a technical and technological base in case the deployment of a multi-layered missile defense system would be necessary."

These “symmetric” defense responses largely revolved around developing a ground-based missile defense, and a space-based defense. However, it’s also important to note that the Soviet ministries proposing these measures were largely repackaging projects that they already had on the books, rather than creating entirely new systems from scratch, and that in any case no development to the point of deployment was considered for at least a decade. Furthermore, Soviet ministries involved in defense projects were confident in developing “asymmetric” responses to SDI (ie, mechanisms for allowing ICBMs to bypass SDI defenses). 

Ultimately, as stated by Pavel Podvig, an independent analyst on Soviet and Russian nuclear forces:

”The new evidence on the Soviet response to SDI largely corroborates the prevailing view that the Soviet Union eventually realized that this program does not present a danger to its security, for it could be relatively easily countered with simple and effective countermeasures. The evidence also helps answer some important questions about the concerns that the Soviet Union had about the U.S. program, the reasoning behind the choices that the Soviet leadership made, and the process that led to those choices.

So SDI does not seem to have greatly altered Soviet military spending. 

Which is not to say that the Soviet government did not care about SDI! The key difference is that it is not that SDI caused a new round of massive military spending, but that there was the fear that it and similar programs might at a time when Gorbachev was already committed to lowering defense expenditures. It clearly was a major item in arms control negotiations between the US and Soviet Union, most notably in the Reykjavik Summit in October of 1986: Gorbachev offered massive reductions in nuclear weapons if Reagan would agree to scrap deployment of (the then-nonexistent) SDI. Reagan refused, but offered to share the technology with the Soviet Union, which Gorbachev was suspicious about (“You don’t even want to share petroleum equipment, automatic machine tools, or equipment for dairies, while sharing SDI would be a second American revolution.”). The end result was that both parties walked away without any agreement. As Reagan noted: “Gorbachev is adamant we must cave in our SDI – well, this will be a case of an irresistible force meeting an immovable object.”

SDI played a major role in US-Soviet arms control negotiations in the 1980s, but it was more of a complicating factor, rather than a decisive factor – if anything it made coming to a comprehensive arms control agreement more difficult. 

Now, I’d like to turn to the Soviet economy and its role in the Soviet collapse.

Part II

Now, the Soviet economy was facing difficulties by the time that Gorbachev became General Secretary in 1985. The Soviet budget had been running small deficits since the 1970s, defense expenditures made up some 18 percent of national income (although other estimates concluded that something like 20-40 percent of the economy was involved in military production), and investment in consumer goods industries and social programs was lagging. 

Khrushchev had considered shifting resources from defense to the consumer sector, but Brezhnev boosted military spending. When Gorbachev came to power, he was willing to cut back on military expenditures in order to free up capital for the civilian economy, although he and those around him would differ on relative priorities and on the need for introducing market mechanisms. However, ultimately, considering the extremely low pay of the largely conscript Soviet military forces, any major savings would have to be made in cutting military industrial production. While all of these debates occurred with SDI on the table, they also occurred with increased Western technological sophistication, stagnating civilian living standards, and a costly and bloody Soviet intervention in Afghanistan. 

By 1988, Gorbachev pushed the economic shift forward with a declaration of unilateral defense spending cuts, personnel cuts of half a million, a planned withdrawal of Soviet military forces from Warsaw Pact states, and a “de-ideologization of interstate relations” in a speech to the UN on December 7. 

Ironically, it was Gorbachev’s attempt to shift the Soviet economy that led to the increasing fiscal instability of the regime. In order to refocus and modernize industrial production, the Soviet Union needed to import new machine tools from abroad. An increase of importation of machine tools, coupled with a fall in international oil revenues (from 30.9 billion rubles in 1984 to 20.7 billion rubles in 1988) caused a massive increase in the deficit: from some 17-18 billion rubles in 1985 to 48-50 billion rubles in 1986, and rising. This was also coupled by a fall in domestic governmental revenue, as Gorbachev’s anti-alcohol campaign cut sales receipts (a Soviet version of a sales tax) from 103 billion rubles in 1983-1984 to 91.5 billion rubles in 1986. The deficit continued to climb, reaching an estimated 120 billion rubles in 1989 (or 10-12 percent of Soviet GNP). By 1990, no one really knew how large the deficit was in reality, and with increasing political reforms giving greater sovereignty to the Soviet Republics, some three fourths of tax collections were withheld from the center by the Republican governments, leading to an effective bankruptcy of the Soviet government. The Soviet government responded to these deficits by printing money, which in turn caused a sharp rise in inflation, an increased scarcity in goods, and a related decline in living standards. Glastnost (greater media openness) meant that increasingly the government was forced to admit the scale of the economic crisis, and the public was very well aware of the problem.

As economist Marshall Goldman notes: 

”Gorbachev’s well-intended but misguided economic strategy was in itself enough to cripple any chance to bring about the economic revitalization he wanted to badly. But the macroeconomic implications of his budget deficit eventually came to have their own impact. Whatever their commitment to socialist economic planning, Soviet officials by 1989 and certainly by 1990 belatedly came to understand that macroeconomics and budget deficits, particularly large ones, do matter. As Gorbachev himself admitted in an October 19, 1990, speech to the Supreme Soviet of the USSR, “We lost control over the financial situation in the country. This was our most serious mistake in the years of perestroika…Achieving a balanced budget today is the number one task and the most important one.” “

In 1990, the Shatalin Plan was proposed, to further cut foreign aid, military expenditures, and the KGB budget in an effort to reign in the deficit and inflation. Other proposed plans called for the cutting of food subsidies. Ultimately, Gorbachev was unable to make a firm decision on any of these proposals, and so the economic situation continued to deteriorate. Confidence in Gorbachev, both from reformers and from conservative hard-liners, fell, until the events of 1991 caused Gorbachev to lose complete control of political events, and ultimately caused the end of the government he stood at the top of.

In summary: SDI, both in terms of its US program and the Soviet response, were more based on research and proposals than on hard spending on a new arms race. The role that SDI played in Soviet concerns over military expenditures was not negligible, but is debatable and probably not decisive. SDI became a major sticking point in US-Soviet arms negotiations by 1986, but if anything hindered agreements concluding the Cold War. By the time Gorbachev assumed power in 1985, the USSR was in economic and social stagnation, and was spending ever more of its national income on defense that was not keeping up with technological advances in the West. Nevertheless, the cuts to military spending that did occur were largely unilateral moves on Gorbachev’s part as a key in his plans for economic, political and social restructuring of the USSR. Overall, it was these reforms, their economic mismanagement, Gorbachev’s often hands-off and indecisive leadership style, and increased social, ethnic and political tensions that resulted which caused the fall of the USSR.

Sources:

Reagan, Ronald. “Address to the Nation on Defense and National Security”. March 23, 1983. Reagan Presidential Library. Link here

Gorbachev, Mikhail. “Speech to the U.N.” 43rd General Assembly Session, December 7, 1988. Link here

Broder, John M. “'Star Wars' First Phase Cost Put at $170 Billion : System Would Intercept Only 16% of Soviet Missiles, Report of 3 Senate Democrats Says” LA Times, June 12, 1988. Link here

“Cost of Missile Defense Put at $70 Billion by 1993”. NY Times. February 12, 1985. Archive link here

Podvig, Pavel. “Did Star Wars Help End the Cold War? Soviet Response to the SDI Program. Russian Nuclear Forces Project, Working Paper. March 17, 2013. Link here

Goldman, Marshall. What Went Wrong with Perestroika. W.W. Norton & Co., 1992

Hoffman, David. The Dead Hand: The Untold Story of the Cold War Arms Race and its Dangerous Legacy. Knopf Doubleday Publishing Group, Sept. 22, 2009

Odom, William E. The Collapse of the Soviet Military. Yale University Press, 2000

Now for some updated thoughts. 

There's a more charitable reading of Reagan and the end of the Cold War, namely that relations were noticeably, massively thawed when he left office in January 1989 than when he arrived in 1981. As US president he deserves some credit for this, because he could have kept a hard line all the way throughout if he had wanted (and he did get public criticism from parts the political right for warming US-Soviet relations). He developed a good relationship with Gorbachev, and was in continous conversation with the Soviet leader, even when individual summits like the one at Reykjavik ended up not going anywhere. Reagan and Gorbachev signed the Intermediate Range Nuclear Forces Treaty in 1987, which was the first arms control treaty to cover an entire class of nuclear weapon ( as of 2019 it's no longer in force). 

But even here, there are some caveats. Most of the events ending the Cold War and the dissolution of the USSR happened after Reagan left office in 1989. George H.W. Bush, although he had been Reagan's Vice President, rather famously put US-Soviet relations on pause for a strategic rethink: he actually had to come around to developing his own trust of Gorbachev, it's not something that was an automatic given. 

And at the end of the day, Gorbachev was always in the driver's seat for Soviet actions. The Soviet withdrawal from Afghanistan, cutback in support for friendly regimes in Africa, Southeast Asia and Latin America, and the withdrawal of troops from East Europe were his initiatives, not something forced on him. The drawdown in Eastern Europe was actually a surprise (he announced it at the UN in 1988), and Americans didn't even have advance notice of it. 

And lastly, "the end of the Cold War" didn't originally mean "the fall of the Soviet Union". Bush and Gorbachev jointly declared the Cold War over at the Malta Summit in December 1989, a good two years before the USSR's dissolution. The events that led to that dissolution in 1989-1991 were internal Soviet political events, and at least up to August 1991 Bush himself advocated for a continued Soviet Union with Gorbachev leading it. It's only once the USSR fell that its dissolution began to be associated to the "real" end of the Cold War, and the US winning it (and Bush himself is responsible for making this association in his 1992 State of the Union speech).

So even in this charitable interpretation of "Reagan ended the Cold War", it tends to assume that the 1991 dissolution and not the 1989 Malta Summit is the end point, but also that all the major events happened because of Reagan, and that even on the American side Bush was just continuing or caretaking those policies, when he had to come around to warm US-Soviet relations on his own, and make a number of very critical decisions.