That’s actually a relatively healthy number for the president compared with his national approval rating (around 41 percent), but it could be a conservative estimate, and voters’ views might be unusually entrenched, given the stability of his ratings.

Many presidents have won re-election after a midterm drubbing, and Mr. Trump’s approval rating isn’t particularly far from what he may wind up needing, given the G.O.P.’s current advantage in the Electoral College. But the wall has not helped so far.

Data from the Fox News Voter Analysis of the midterms, a new competitor to the traditional exit polls, indicated that a majority of voters opposed the wall in states worth nearly 400 electoral votes, including in several states where the president’s approval rating was above water in the poll, like Ohio and Florida.

There is tentative evidence in the Fox data that the wall is particularly unpopular in the relatively white and rural West, but somewhat more popular, at least with respect to the president’s approval rating, in the Northeast and inland South. This would follow a familiar pattern in American politics: It mirrors the president’s support in the presidential primary and tracks with longstanding measures of racial resentment.

Even so, the wall still isn’t popular in Michigan or Pennsylvania, important battleground states. And voters in Ohio, a politically similar state, opposed the wall.

None of these polls account for the government shutdown, but attitudes about it are unlikely to be fluid given the lack of movement on the wall during the Trump presidency. Tying the issue to an unpopular shutdown seems particularly unlikely to help and, historically, voters tend to drift against the policy preferences of the president’s party. That’s already evident on immigration more generally: Polls show voters more supportive of immigration than ever before.

For all of the focus on the president’s base over the last two years, there is not much reason to think that the base, alone, is enough for the president to win re-election in a one-on-one race against a viable Democratic candidate. This could change. It has before. But with the midterms over, this is now the central political challenge facing the president. By that measure, it’s hard to see where a shutdown over the wall fits in.