Even before President Trump spoke to the nation around 4pm today, the Democrats outright dismissed his proposals. Nancy Pelosi called them a non-starter. Yet, simply by making what look like concessions, the president did the first somewhat sensible thing since the government shut down before Christmas. Some of what he offered is of value to certain Democratic constituencies. Some of the money he's approved would go for border initiatives that the Democrats actually support. So, the Democrats need to respond with something beyond intractable rejection or the stupid speech will be a success.
There is some potential that the Democrats will begin a splinter, even if the leadership is unlikely to be moved. Mitch McConnell intends to bring a bill up for a Vote based on the president's proposals and the first test will be a cloture vote requiring sixty senators (or seven Democrats). If successful, that would only authorize the Senate to begin a debate and to vote on amendments. They would have to pass the sixty vote threshold on the backend, too.
Therefore, the Dems could allow the process to start while reserving the right to block a vote on final passage if the bill isn't to their liking. That might be a tempting tack to take, but it would probably just needlessly string out the government shutdown.
Either way, the House would have to sign off on the bill, so they might a well let the House take the lead. Pelosi is firm on not beginning a negotiation until the government is reopened, but she can pass new proposals almost at will. That's certainly how she will proceed. It will keep her more vulnerable members from revolting if they can continually cast votes to reopen the government, and some of those bills can incorporate things Trump just agreed to in his speech.
Trump succeeded in applying some new pressure but it is very unlikely to break the impasse or even lead to any major concessions.