The no-wall deal that ended the government shutdown was a bitter pill for voters who sent President Trump to the White House, but they say they’re still backing him as the border security fight with Democrats enters Round Two.

Even as conservative pundits blasted the president for “caving” to House Speaker Nancy Pelosi, voters contacted by The Washington Times said they aren’t holding the setback against Mr. Trump.

“I haven’t lost any confidence in him or faith in him,” said Brian McKenna, 73, a retired police officer in Stuart, Florida. “I’m dumbfounded that people in this country don’t see the problem down at the border. The whole issue is about somebody breaking the law and the Democrats defending it.”

While Mr. Trump’s base still wants a wall or barrier on the U.S.-Mexico border, their support for the shutdown had slowly eroded as the impasse dragged on for 35 days.

“He lost on this, but his hands were kind of tied,” said Gus Lester, 66, a disappointed Trump supporter on Maryland’s rural Eastern Shore who said has not yet given up on the president or his border wall.

Mr. Lester, a lifelong Democrat and longtime Carpenters Union president who crossed party lines to vote for Mr. Trump in 2016, said the government shutdown had to end.

“We still need the wall,” he said. “Let’s see what happens [three weeks] from now.”

The president’s agreement with Democrats calls for House-Senate negotiations on funding for the Department of Homeland Security and border security, with a deadline of Feb. 15. Mr. Trump has warned that he could declare a national emergency if he doesn’t get some amount of money for 230 miles of barriers along the southern border.

Said Mr. McKenna, “Three weeks from now he’s going to have do something if [Democrats] won’t negotiate.”

The president weathered harsh criticism from conservative pundits and media personalities for failing to secure funding for a wall. Fox Business host Lou Dobbs, usually a supporter of Mr. Trump, said of the agreement to end the shutdown, “Illegal immigrants are surely pleased at the prospect they may soon jump to the front of the line while legal immigrants aren’t even part of the discussion in the nation’s capital.”

Conservative activist Ann Coulter called Mr. Trump a “wimp” and lamented, “It’s crazy that I expect a president to keep the promise he made every day for 18 months.”

A senior White House official said dismissively of Ms. Coulter, “The president of the United States isn’t going to react to what somebody is screaming on Twitter.”

The official said that Mr. Trump has succeeded in “elevating border security in the conversation,” and that negotiations over the next three weeks will target at least 30 House Democrats serving in districts that Mr. Trump won in 2016, including 22 freshman lawmakers.

“Nancy [Pelosi] has been totally hijacked by the left wing,” the senior official said. “He [Mr. Trump] hears from Democrats. They know their leadership isn’t representing their districts.”

Given those dynamics, the senior official said the president’s approach will be, “OK, government’s open. Let’s talk.”

A letter written by first-term Rep. Elaine Luria, Virginia Democrat, and signed by more than 30 House Democrats last week called for a vote on the president’s border-security proposal after government reopened. White House deputy press secretary Hogan Gidley said “the dam was beginning to break on the Democrat side and we could get something accomplished. That’s why the president decided to open the government.”

But others noted that six GOP senators had voted last week for the Democrats‘ plan to reopen government without wall funding, and that Mr. Trump ended the shutdown shortly after a shortage of air-traffic controllers Friday delayed hundreds of flights at several major airports.

Republican strategist Ford O’Connell said Mr. Trump’s support will remain intact as long as he keeps fighting for a border wall, whether in negotiations with Congress or in a court battle over a national emergency declaration.

“He might be taking a short-term hit with his base but really you have to withhold judgment until Feb. 15,” he said. “As long as Trump is seen as committed to border security and physical barriers at the southern border, it is not going to hurt him in the grand scheme of things.”

The deal, while chalked up as a win for Mrs. Pelosi, puts the burden on Democrats offer a border-security plan.

“Democrats say over-and-over they care about border security at the U.S. southern border. Now we are going to find out just how serious they are,” Mr. O’Connell said. “Trump is making a good faith offer and trying to govern through the legislative process, and if Congress doesn’t pick up the ball I wouldn’t be surprised to see him declare a national emergency.”

Tea Party Patriots founder Jenny Beth Martin, who met with the president and other conservative leaders at the White House last week, said the president’s base is blaming Mrs. Pelosi for refusing to negotiate on border security.

“The whole point of this exercise is to secure the border,” she said on Fox & Friends. “What President Trump has promised, she’s not willing to negotiate on.”

She said Mr. Trump’s supporters are standing with him in spite of his earlier offer of protections for nearly 1 million illegal immigrants and people with Temporary Protected Status.

“He’s been willing to go further than some people may even want him to do who are his supporters,” Ms. Martin said. “But they understand. They’re trying to give him the leeway to negotiate. She [Mrs. Pelosi] hasn’t been willing to do that at all.”

Hispanic faith leaders who met with Mr. Trump at the White House Friday just after the shutdown ended told him they support his efforts on border security.

“You’ve done a very good job being the leader we need you to be,” Pastor Ramiro Pena of Christ the King Baptist Church in Waco, Texas, told the president. “We want to say thank you for doing everything you can get aid in a humanitarian way for this crisis where we need it most, securing our borders, and providing for law enforcement and the Border Patrol everything they need.

He added, “One thing you have not done, Mr. President, is manufacture a humanitarian crisis. It’s real. I’m grateful that you have shown leadership to recognize it, to call it out, and actually do something about it.”