If Mr. Trump has animated a powerful national campaign against him, Democratic candidates have largely avoided engaging the president personally in the closing days of the election, instead hewing close to a few favored issues like health care.

At a Saturday morning rally, Representative Ben Ray Luján of New Mexico, the head of the Democrats’ campaign committee in the House, drummed home the party’s ethos of ignoring Mr. Trump while riding the backlash against him.

“We don’t really have to even talk about this president — he’s going to do all the talking about himself, for himself,” Mr. Luján said, addressing volunteers in Los Lunas, where Democrats are making a push to pick up an open House seat. “I want you to concentrate on families here in New Mexico.”

But Senator Martin Henrich, appearing beside Mr. Luján and Xochitl Torres Small, a water-use lawyer who is the Democratic nominee for Congress, cast the election in dire terms familiar to anxious Democrats across the country. “This is a battle for who we are as a nation,” said Mr. Heinrich, who is expected to win re-election easily on Tuesday.