As negotiations unfolded between Senate Republicans and Dr. Blasey about her testifying before Congress, some party officials were growing nervous about how President Trump and the largely male slate of Senate candidates were handling what could turn out to be political dynamite. Mr. Trump, breaking from a period of relative restraint, said on Friday that Dr. Blasey would surely have reported her alleged assault to authorities if it “was as bad as she says,” and a handful of Senate Republican candidates have minimized or dismissed her claims.

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In Missouri, sexual misconduct is still fresh on the minds of voters after the scandal that drove former Gov. Eric Greitens from office this year. And being tone-deaf on the treatment of women memorably hurt Ms. McCaskill’s Senate opponent in 2012, Todd Akin, who led in some polls until he said that women who suffer “legitimate rape” rarely get pregnant.

“If this is actually on TV next week, and women see she is telling the truth, that could be very, very persuasive,” former Gov. Jay Nixon of Missouri, a Democrat, said.

John Danforth, the former Republican senator from Missouri who was Justice Clarence Thomas’s chief Senate patron during his 1991 confirmation hearings, urged his former colleagues to have an outside counsel question Dr. Blasey when she appears before the Judiciary Committee.

“Members of the Senate, especially in the #MeToo era, have got to walk on eggshells,” Mr. Danforth said.

As for the current Missouri Senate campaign, Mr. Danforth, who is something of a mentor to Mr. Hawley, acknowledged that the “jurisprudential issues” of policy and law had “been overtaken by this particular episode.”