Ken Starr and Newt Gingrich did Clinton a big favor. They were nakedly partisan attention whores and extremely unlikeable people - Gingrich in particular is one of the most repulsive figures in modern political history. I have my doubts Clinton would have been reelected without Gingrich screwing up the optics of 1995 and 1996 as badly as he did. Starr handled everything about the Starr report horribly - the decision to fully release a report that was so incredibly crude was a terrible mistake, because it made a media circus focused on cigars and blue dresses, which made late night comedy fodder and which disgusted the average voter. It also shifted the focus to being about morals, which meant when one leading Republican after another was revealed as an adulterer, the public saw everyone as being the same, or the Republicans as just being hypocrites.
Nancy Pelosi is not a media whore and I think she will handle this judiciously and as intelligently as possible, but beyond the Senate roadblocking any real investigation, there's also the amount of apathy and partisanship out there now. Many people will automatically believe Trump because they adore Trump. Many others simply don't care. They see all politicians as the same, and they are more worried about not being able to pay their bills.
I think Pelosi knows that deep down this isn't going to do anything, but she was slowly but surely put into a position where there was going to be a huge cost for the Democrats in 2020 in terms of voter turnout and donations if she didn't act.