Because it would show that he's compromised by a foreign power having salacious sex tapes on him. He could not be relied upon to reasonably fulfill the duties of the office.
It would be far worse than Bill Clinton getting a BJ from a naive nobody intern in 1998.
I think the odds of it are reasonably high too. The British spy is somone who's verified by the FBI and CIA to be reliable and they've used intel from him in the past. They also verified parts of the dossier.
Secondly Trump himself in one of his business books years ago bragged about basically going to Europe for sex (he cites this as one of the reasons why he went bankrupt the first time because he "took his eyes off the ball").
Really, it doesn't take much to believe that Russia knowing that he's a billionaire tapped his hotel room. Likely all high profile visitors there are tapped.
If all the things he has done before don't prove that he is unfit for office, nothing will. That he is unfit was clear long befor the election. I don't see how this completely unacceptable action of a president is any different from the multitude of other unacceptable actions he took the past year.
He will just claim fake news, fire someone and then have his next unacceptable gaffe so people forget about it.
No actually that is an impeachable offence.
You would be effectively black mailed into doing things that might go against national interest by a foreign power. That is a big, big no, no. That is like 100x worse than Watergate that got Nixon booted. If any of that ever came to light he'd be done.
You can't "fake news" your way out of that. In any case I think that who Trump train will eventually collapse on itself. His whole appeal was based around being different and "lets give the crazy guy a shot", but generally speaking people are still dissatisfied with Washington, and IMO they will still be in 3 years. The novelty of him will wear off quick. He's all talk and his ideas don't bring any meaningful change to anyone's life, nor does he have any ability to get any reasonable things done (being a complete novice in politics doesn't help). Policy wise he's really at his core just another "give more tax breaks to rich people, cut health care for poor people" Republican.
All that "running things like a business" was a load of shit. Governing is nothing like running a business.