Last week, speaking at Northwestern University in Evanston, Ill., Mr. Obama declared that while he was not up for re-election, his “policies are on the ballot.” Immediately, Republicans pounced, putting the clip in videos to link their rivals to the president. Democrats winced, and David Axelrod, the longtime Obama adviser, acknowledged Sunday on NBC’s “Meet the Press” that the remark was “a mistake.”

A succession of domestic and foreign crises, along with some self-inflicted wounds, has badly tarnished Mr. Obama. And that is on top of the history of the president’s party doing poorly in midterm elections.

But Mr. Obama is no ordinary president in the eyes of his supporters, who believe he has permanently reshaped the makeup of his party and the way Democrats ought to go about winning elections. And his diminished role in this year’s campaign has set off the first stirrings of a debate sure to grow louder if Democrats lose the Senate next month.

“Every campaign has got to figure out — and this is true in this election, and it’s going to be true for every election going forward for Democrats for as far as the eye can see — is which Democrats are only going to be able to win if they turn out enough of the Obama coalition, whether we’re in a midterm or a presidential,” said Dan Pfeiffer, a senior adviser to the president who has been with him since his 2008 campaign. “And the campaigns and the candidates are working through how best to go about doing that.”

Mr. Obama’s aides say they understand that candidates will make their own decisions as it relates to using the president. But a central tenet of Mr. Obama’s formula for success is that Democrats win by motivating core voters to turn out as much as they do by persuading swing voters — and that nobody can rouse the base like Mr. Obama. So for candidates to distance themselves from the president, or even disparage him, is to ignore a potential path to victory.