When Sen. Tim Kaine was vetted in 2016 to be Hillary Clinton’s running mate, the Virginia Democrat was prodded about everything from his financial dealings to the health of his ticker.

But Mr. Kaine wasn’t specifically asked if he ever gallivanted around in blackface.

“I was asked virtually everything you might imagine, but I was not asked that,” Mr. Kaine told The Washington Times on Thursday. “I have been asked that in the last few days, and the answer is ‘no.’ “

As the party looks for candidates for 2020 and beyond, it’s trying to avoid a repeat of the mess going on in Virginia, where the top three elected leaders — all Democrats — have been stung by race and rape allegations.

Bret Niles, chairman of the Linn County Iowa Democrats, said he wouldn’t be surprised if candidates visiting the early primary states don’t get asked to weigh in.

As for the candidates themselves, he said he hoped they would be forthcoming on their own, and wouldn’t need a scandal to make them admit to any politically damaging shortcomings.

“I think there are a lot people wanting to make sure we have a presidential nominee with a clean slate, or nothing in their closets, that would impact the campaign or their term in office,” he said. “However, it’s hard to envision any of the current candidates with anything similar to what all came out on Trump, even during the campaign.”

Virginia Democrats may have felt the same about their governor, lieutenant governor and attorney general, the first two elected and the latter re-elected in a sweep of the state’s top offices in 2017.

But over the last week, Gov. Ralph Northam has had to answer for a photo of someone in blackface and another in a Ku Klux Klan costume that appeared on his 1984 medical school yearbook page. He denied he was in the photo, but admitted to darkening his face at a different time to appear black for a dance contest.

Lt. Gov. Justin Fairfax, who would take over should Mr. Northam resign, was then publicly accused of raping a woman when they were both attending the Democratic National Convention in Boston in 2004.

And state Attorney General Mark Herring, who had called on Mr. Northam to resign after the blackface photo emerged, admitted this week to his own blackface incident in 1980.

The chaos has garnered wall-to-wall cable news coverage and has spilled over to Capitol Hill, where House Speaker Nancy Pelosi maintained Thursday that it doesn’t taint the party’s prospects at the national level.

“No, it does not,” the California Democrat told reporters.

Mrs. Pelosi said Virginia “will resolve” its own issues and said it was “sad” the party was suffering because it had “some very talented leaders.” She did not say who she meant.

But Democrats said they do expect 2020 candidates will be asked to weigh in.

Many in the field have called for Mr. Northam to step down.

Sens. Kamala D. Harris of California and Kirsten Gillibrand of New York on Thursday became the first major contenders in the field to say Mr. Fairfax’s accuser is credible and to call for her allegations against the lieutenant governor to be investigated.

Mr. Niles said he wouldn’t be surprised if Ms. Harris or Sen. Cory Booker of New Jersey is asked about the mess in Virginia when they travel to Iowa this weekend.

He said Democrats still feel burned by John Edwards, who convinced voters during his 2008 presidential bid not to believe reports from the National Enquirer that he had a love child with a mistress. The reports proved to be true.

Kathleen Sullivan, a member of the Democratic National Committee from New Hampshire, said the stakes in 2020 are so high that it is incumbent upon Democrats to do everything in their power to put forward a scandal-free presidential nominee.

“I do think that Democratic voters will be more attuned to issues of race, gender, harassment, but not just because of Virginia, but also because of Donald Trump, [Steve] King, and others in the Republican Party who have histories that are problematic, to say the least,” Mrs. Sullivan said.

Mr. Kaine, who early on called for Mr. Northam to step down but has not embraced calls for the other two to go, declined to weigh in on how his state’s mess might affect national candidates.

“I am not really thinking about the politics right now,” Mr. Kaine said. “There are principles involved in each of these cases — racial sensitivity, treatment of women — and we have to kind of get the principles right, then there will be an opportunity to figure out the politics later.”

• Stephen Dinan contributed to this report.

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