The FBI investigates
1) Federal Crimes
2) Kidnapping (only because of the presumption that the victim may have been moved across state lines)
If Kavanaugh jumped on stage at a Kanye West concert and groped his backup dancers in front of 40,000 people the FBI should refuse to investigate, right?
...because sexual assault is a state crime, no?
They wouldn't be doing the investigation to decide if Kavanaugh should be prosecuted in Court. That is, indeed, not their job in this case.
But they do background checks on nominees, and can be asked to do so by the White House. That is what happened with Clarence Thomas, and no one then argued workplace sexual harassment was a federal crime.