“The only thing Hillary can do, I think, is get out there in front of reporters and take five hours of questions — if that’s what it takes — until people understand her, at least, and hopefully believe her,” Ms. Wyman said. “And we have to have people who are talented, independent experts on classified secrets defending her on television, rather than who we have now.”

Criticisms about Mrs. Clinton also threaten to complicate the political fortunes of other Democratic candidates over the next 15 months, should they find themselves having to defend her instead of focusing on issues and going on the offensive against Republicans.

“I obviously think the sooner Secretary Clinton can get this behind her, the better,” said Jason Kander, the secretary of state in Missouri, who is running for the Senate next year against the Republican incumbent, Roy Blunt.

Mrs. Clinton has stressed that her use of only a private email account was legal, and she is not a target of a federal investigation, which is focused on assessing any security breaches.

Still, with hundreds of Democratic National Committee members gathering in Minneapolis this week for their summer meeting — at which Mrs. Clinton and other candidates will speak on Friday — several said in interviews that they would be listening closely to her remarks on the email issue.

In Kentucky, where a hard-fought governor’s race will be decided in November, some Democrats are privately fretting that Mrs. Clinton’s problems could damage the party’s brand. The departing Democratic governor, Steven L. Beshear, said he believed that Republican attacks over Mrs. Clinton’s email use would continue through the fall and beyond — and that the Clinton campaign had to devise a strategy of more effective countermoves.

“Honestly, at this point there isn’t any great way to handle it,” Mr. Beshear said. “While Hillary Clinton has been straight up from the very beginning, the Republicans will not let the issue die, and they will conduct as many witch hunts as possible. She will really have to start addressing those attacks as the campaign becomes more active.”