Prominent Democrats, including some women thought to be on Joe Biden’s running-mate shortlist, have lined up behind the former vice president in the month since Tara Reade, a former Senate staffer, made public an allegation he sexually assaulted her once in the 1990s.

Sens. Amy Klobuchar of Minnesota and Kamala Harris of California and former Georgia House Minority Leader Stacey Abrams all have said that they respect women making such allegations and take them seriously. But they haven’t wavered in their support for Mr. Biden,...