“I do think that using Trump imagery is triggering for black people,” Ms. Shropshire said. “We don’t use it in our advertising.”

Democratic presidential candidates have not settled on how to deal with the man they all hope to replace. While five of the party’s 20 candidates called Mr. Trump a racist during this week’s debates in Detroit, the party’s private polling shows that affixing that label to him is not the most effective way to peel support away from the president.

A poll conducted in June for the Center for American Progress Action Fund, a political arm of the progressive think tank, offered voters six derogatory descriptions of Mr. Trump: ineffective, false promises, for the rich, divisive, corrupt and racist. Among voters surveyed, the ineffective label moved the most voters toward a generic Democratic candidate; the racist label moved the fewest.

Among black voters, the poll found that calling Mr. Trump a racist did not move support to Democrats. Calling him ineffective did.

“This isn’t about just speaking to the obvious, that our president is a racist, it has to be about how are you connected to the struggle of our communities,” Senator Cory Booker of New Jersey, one of the five candidates who called Mr. Trump racist during the debates, said in an interview on Friday. “I’ve heard that line from candidates before, but not followed by the kind of from-the-heart talk that I think is really important if there is going to be trust that the next leader really feels us, understands communities of color.”

But Lt. Gov. Mandela Barnes of Wisconsin, a state that saw one of the largest drops in African-American turnout between 2012 and 2016, warned his party that the eventual standard-bearer must speak unambiguously about the president’s conduct if he or she wants to energize black voters.

“We still have to have a candidate who won’t be afraid to stand up to him and call him out,” said Mr. Barnes, who is black. He added that the presence of an African-American candidate on the ticket “certainly would be helpful.”