Kristol was Quayles Chief of Staff.....and my guess is that he sabotaged him a lot....just a guess.
1. Convinced Quayle to make that "Murphy Brown" speech, which made Quayle a laughingstock on the late night shows and in the press. (Source: Bill Kristol himself, joking about this to Brit Hume)
2. Bill Kristol convinced Quayle to make a speech critical of Bush 41's foreign policy, right before the kick-off of the 92 campaign. It effectively isolated Quayle from the rest of the administration. (Source, Quayle's book Standing Firm)
3. Kristol was distrusted by the Bush White House, and James Baker wouldn't allow Kristol to sit in on campaign meetings, because he leaked everything to the press. Quayle says that this was because Baker was intimidated by Kristol's intellect (HA!) but that sounds to me like a tale Kristol told Quayle. (Source..Quayle's book).
I have been digging stuff up on Kristol since 2000. He is nothing but a disruptive force and should never be allowed in any governmental position.