This falls under "Durable Myths of the Mafia":

MYTH: Joe Kennedy, JFK's father, was a big-time rum runner during Prohibition. He developed ties with organized crime that he continued to use after Repeal whenever it suited him. In 1960 he contacted Sam Giancana and made him an offer he couldn't refuse: If The Outfit and their ally, the Teamsters, helped his son carry Illinois in the Presidential election, he, Joe Kennedy, would guarantee that the Kennedy Administration would keep hands off the Mafia. Giancana fell for it. But then JFK appointed his brother, Bobby, Attorney General. Bobby declared war on The Mob and on Jimmy Hoffa, Teamsters president. So, The Mob killed JFK as revenge and to get the government off The Mob's back.

REALITY: Joe Kennedy never was a rum-runner during Prohibition. This myth got started because, in 1932, as Prohibition was about to be repealed, he formed a partnership, Somerset Imports, and in '33, after Repeal, started bringing in several brands of scotch and gin from the UK, legally. His partner in Somerset Imports: James Roosevelt, the President's son.

Giancana had no motivation at all to help JFK. In '59, he was subpoenaed to appear before the Senate Permanent Investigation Subcommittee, which was holding hearings on OC influence over labor unions. JFK was a Democrat on the subcomittee, Bobby was chief counsel. Giancana took the Fifth Amendment each time he was asked a question, and gave a little laugh each time. "Why are you giggling?" Bobby asked him, in front of the newsreel cameras, "I thought only little girls giggle." You don't say that to a Mob boss in front of the world and expect him to support your brother for President.

That same hearing also launched the long-running war of the Kennedys against the Teamsters and Hoffa. "Why don't you stop hiding behind the Fifth Amendment and answer questions truthfully," JFK shouted at Hoffa. Bobby and Hoffa almost came to blows at that hearing. The Teamsters supported Richard Nixon for President in1960.

JFK didn't need The Outfit or the Teamsters to carry Illinois because it was promised to him--and delivered--by his friend and ally Richard Daley, Chicago's all-powerful Mayor, via the time-honored tradition of having the dead vote early and often. Nixon was comfortably ahead by about 50k votes until about 11 pm on election night, when a huge block of votes for JFK came in from Cook County. JFK took Illinois by 5500 votes.

Yes, there was crime in the 1960 election: "Grand Theft - Illinois." But, it's so much more exciting and appealing to believe that the Almighty Mafia elected, and then killed, JFK.