“People are even more sensitive about this, and listening, and being sure that women are heard,” Mr. Bennett said. “But on the other hand, there were very serious allegations against Donald Trump and he was elected president. There were very serious allegations against Brett Kavanaugh, and he was confirmed,” he said, referring to the Supreme Court justice.

He called for the Senate to conduct a “very thorough investigation” of the claims against Mr. Cain as part of any prospective confirmation hearing.

“The allegations are very serious,” Mr. Bennett said. “If you have one complaint, it’s a ‘he said, she said’ situation. If you have three or four complaints — the expression is where there’s smoke, there’s fire.”

Mr. Bennett said he was unsure if Ms. Kraushaar would participate in any congressional investigation, explaining that she had recently retired from a career in the federal government and was considering moving away from the Washington area.

It is an uncomfortably familiar spot for Senate Republicans, who were confronted with a similar dilemma six months ago after Mr. Trump nominated Mr. Kavanaugh for the Supreme Court. Mr. Kavanaugh, who was accused by Christine Blasey Ford of sexual misconduct, was eventually confirmed after an excruciating public hearing in which Dr. Blasey testified that he pinned her down and groped her at a party when she was in high school in the 1980s. Mr. Kavanaugh disputed the claims and denied he ever assaulted her.

The choice of Mr. Cain was viewed as especially perplexing in the immediate aftermath of the selection of Mr. Moore, who is dealing with his own accusations of ethical lapses related to his 2011 divorce. Since Mr. Moore was chosen, it has emerged that he owes $75,000 in federal taxes, interest and penalties stemming from his attempt to deduct child support payments, which are not eligible for a tax break. He was held in contempt of court in 2013 for failing to pay more than $330,000 for spousal support, child support and lawyer’s fees to his former wife, Allison Moore.

While Mr. Cain was defiant amid the downfall of his 2012 presidential bid, the toll that a process can take on a nominee and his family was on display on Friday, as Ms. Moore appeared alone in a Virginia court and tearfully pleaded with a judge to keep her divorce records sealed.