What is the story behind the expression "Would you buy a used car from this man?"

I was skeptical that the question "Would you buy a used car from this man?" as a generic expression of doubt about someone's reliability originated as a rhetorical question about Richard Nixon. But an Elephind search gives some credance to the idea. The earliest relevant match for the phrase "buy a used car from" in that newspaper database is from the Tom Henderson, "Convention Sideshow," in the Coronado [California] Eagle and Journal (July 21, 1960) (roughly three months before the 1960 presidential election, in which John Kennedy defeated Nixon):

One of the big questions in the minds of everyone who attended the Democratic National Convention last week in the Los Angeles Sports Arena is, "How the devil are they going to get all those balloons off the ceiling?"

It took a good organization to win the nomination, as was evidenced by the ubiquitous machine of Kennedy as compared to that of the "What, me worry?" kid, Mad Magzine's Alfred E. Neuman, whose placards were seen about the arena proclaiming the motto, "Integrity — Absurdity." Alfred didn't gt a single vote. Even Orval Faubus got half a vote.

Nixon’s picture also came into the scene, a picture of the Vice President at his grimmest over a caption asking, "Would you buy a used car from this man?"

Some notable Republicans attended the Democratic pow wow, but only Mayor Poulson of Los Angeles got as far as the platform. He gave the welcoming address.

So the place of origin of "Would you buy a used car from this man?" as a way to express doubt about a person's trustworthiness and integrity may indeed have been the Democratic National Convention of 1960, where some anonymous Democratic operative used it as a provocative challenge to Richard Nixon's honesty and character.

Interestingly, Republicans attempted to float a similar question about Lyndon Johnson in the 1964 presidential election, but abandoned it when it failed to gain traction with prospective voters, according to a syndicated column by Jack Wilson published on October 28, 1964:

Notice that the Republicans have dropped that line, "Would you buy a used car from LBJ?" They got a backlash—turns out there were more used car dealers than customers.

By 1965, the expression was familiar enough to U.S. audiences to be used widely in nonpolitical contexts. For example, country-western singer Norma Jean Beasler had a hit song in 1965 called "I Wouldn't Buy a Used Car from Him," which reached number 8 on the C&W charts. The song is about an untrustworthy former sweetheart and doesn't seem to allude to Nixon (or Johnson) at all.

When Richard Nixon ran for president in 1968 (against Hubert Humphrey), the old question became popular once more. For several weeks before that election, the Columbia [University, New York] Spectator ran an advertisement for a campaign button that read "I wouldn't buy a used car from either one." Maybe that's how Nixon won.

I might have been very small at the time and I do remember how half the country thought it was very funny when that phrase was applied to Tricky Dicky.

It might not have been coined for him but it was clearly popularised by him - or by his detractors.

The idea did work only because the poor garage guys already had a dreadful reputation but equally, the whole point of plastering it over political posters was that previously it hadn’t appeared on that scale, if at all in those words.

I still have a Christmas card from the period with a friend’s family picture over that slogan, purely because at the time it was so unusual at the time.

The writer and journalist Noel E. Parmentel Jr. originated the phase “Would you buy a used car from this man?” applied to Nixon. See his Wikipedia page for references. Mr. Parmentel is still living as of the date of this note. He served with my father on the USS Chester in WWII.