“There are people who have had to withdraw over the years because you get information out and you question them and the public is focused on it and galvanized by it,” said Senator Amy Klobuchar, Democrat of Minnesota and a member of the Judiciary Committee.

Demand Justice has begun running online ads against three of Mr. Trump’s possible picks and expects to spend more than $1 million on television ads once the president selects a nominee.

One ad targets Judge Amy Coney Barrett, who is on the president’s list of possible justices, for saying that the Affordable Care Act should have been held unconstitutional. Another ad is aimed at Judge Brett Kavanaugh, another possible pick, for saying that a president should be able to “decide whether and when he can be investigated.”

Democrats say they will focus on two main issues, abortion and health care. Mr. Fallon, who worked for President Barack Obama and was a top spokesman for Hillary Clinton’s 2016 presidential campaign, said the challenge will be convincing people that a Trump justice poses a threat to both issues.

“We have to do the work and spend the money to communicate the consequences of what a 5-4 court with a newly installed justice looks like,” Mr. Fallon said.

Democratic lawmakers gathered on the steps of the Supreme Court Thursday morning, flanked by members of progressive groups, to declare their opposition to all of the potential candidates on Mr. Trump’s public list of 25 possible jurists.