Senator Bernie Sanders, independent of Vermont, who ran unsuccessfully for the Democratic presidential nomination in 2016, used a response to the president he streamed live on YouTube to brand Mr. Trump a liar, and ticked down a list of misstatements the president uttered during his address.

“It gives me no pleasure to tell you what most of you already know, and that is that President Trump lies all of the time, and in remarks tonight and in recent weeks regarding immigration and the wall, he continues to lie,” Mr. Sanders said.

He went on to discuss the plight of millions of Americans who lack health insurance, elderly people who have no retirement savings, millions more with crippling student debt and the specter of global climate change, arguing that those challenges were far more pressing than the one posed by illegal immigration.

“We don’t need to create artificial crises,” Mr. Sanders said. “We have enough real crises.”

Mr. Trump is planning a trip to the border in McAllen, Tex., on Thursday, while Democrats plan to bring up a succession of bills this week to reopen shuttered parts of the government, including the Internal Revenue Service, to allow tax refunds to be paid, and the Agriculture Department, to ensure that food assistance and farmer support payments can be made.

The Treasury Department and the I.R.S. will come first, then Agriculture and Interior Department programs, including the national parks, on Thursday, followed by transportation and housing programs on Friday.

“We are going to, daily, urge and take efforts to open up the government,” said Representative Steny H. Hoyer of Maryland, the majority leader. “Whatever problems confront us, they are exacerbated — not relieved — by shutting down government.”

A group of Senate Democrats spent Tuesday evening taking turns on the Senate floor calling on Senator Mitch McConnell of Kentucky, the majority leader, to bring up the House-passed legislation to reopen the government. The dozen Democrats, led by Senator Tim Kaine of Virginia and Senator Jeanne Shaheen of New Hampshire, plan to spotlight the harm from the shutdown on federal workers and those who benefit from government programs.