I don’t know if this would come up in a background check.
Unless this involved assault with a dildo, it is irrelevant.
They are just trying to destroy Kavanaugh.
Well it depends to what depth they are going. It would have definitely come out on a Top Secret or Secret clearance investigation.
The FBI knocks on all the neighbors' and coworkers' doors and asks these kind of questions, among many others. The FBI pretty much is looking for character flaws, unusual behavior, or reason that a person could be suspeptiable to blackmail, reasons such as homosexuality, adultery, or financial stress.
There is nothing here with this new story. Just another stunt.