The rush of Democratic governors and senators, both moderates and progressives, to airports this weekend also showed that they recognized that the outpouring of protesters for the women’s marches on inauguration weekend was only the start of a new movement.

“I think the key for the Democratic Party is to show to a very powerful and dynamic coalition that we are hearing them and connecting to their issues,” said former Representative Tom Perriello, who made a surprise entry into the Virginia governor’s race after Mr. Trump’s election and drove to Dulles International Airport from Richmond, Va., on Saturday to join the protests.

The demands for purity from Democratic leaders are likely to grow only more unequivocal, and standing in solidarity with sign-waving protesters may prove insufficient. A rising chorus of liberals is pressuring senators to filibuster Mr. Trump’s selection for the Supreme Court, which he has said he will announce this week.

“When it comes to the Supreme Court, eight is enough,” said Representative Keith Ellison, a Minnesota Democrat who is one of the top contenders to be the chairman of the Democratic National Committee. “They literally stole a Supreme Court justice from President Obama, so I see no reason why we should facilitate them putting in some right-wing extremist.”

But some Democratic senators, including those facing re-election next year, are uneasy about forming a blockade against Mr. Trump’s pick, as Senate Republicans did last year against Mr. Obama’s nominee, Judge Merrick B. Garland. Doing so could be fruitless, anyway. Mr. Trump has already encouraged Senator Mitch McConnell of Kentucky, the majority leader, to ram through new Senate rules that would allow a Supreme Court pick to be seated with a simple majority of 51 votes — the same rules change Democrats engineered in 2013 to end filibusters of virtually every presidential appointment besides a Supreme Court justice.

“I believe we must fulfill our constitutional responsibility to have a hearing and a vote,” said Senator Amy Klobuchar of Minnesota, a member of the Judiciary Committee, while urging Mr. Trump to choose a nominee “in the mainstream.”