Asked directly if she believed Mr. Clinton should have stepped down at the time, Ms. Gillibrand took a long pause and said, “Yes, I think that is the appropriate response.”

But she also appeared to signal that what is currently considered a fireable offense may have been more often overlooked during the Clinton era.

“Things have changed today, and I think under those circumstances there should be a very different reaction,” Ms. Gillibrand said. “And I think in light of this conversation, we should have a very different conversation about President Trump, and a very different conversation about allegations against him.”