It Depends
JFK could very well have sucessfully outmaneuvered Senator Este Kefauver of TN at the 1956 Democratic National Convention in Chicago to win the VP nomination. After fmr IL Governor Adlai Stevenson, the presidential nominee, decided let the convention delegates determine whom his running would be, JFK and Kefauver had only one day to campaign amongst the delegates. On the second round of balloting, JFK was in the lead and only 15 votes shy of winning the VP nomination.
If JFK had won on the second ballot and become Stevenson's running mate, the danger was that the inevitable loss by the Stevenson-Kennedy ticket to Eisenhower-Nixon can ticket would be blamed, at least in part, on Kennedy--in particular his youth and Catholic religion--and hurt his own plans for a future presidential bid.
As it was, Eisenhower-Nixon won 41 of the then-48 states and over 57% of the popular vote in the 1956 election. Stevenson-Kefauver ran a poor campaign and won 6 southern states, plus Harry Truman's Missouri, and only 42% of the popular vote. Outside the South and a few border states, Eisenhower won every state with 55% of the popular vote or more--including JFK's Massachusetts by whopping 18-point margin (59%-41%). While Joe Kennedy's connections and money employed more liberally on behalf of the Stevenson-Kennedy ticket might have improved the quality of the national Democratic campaign, it is difficult to see JFK's presence on the ticket helping very much in terms of the final popular or electoral vote tally.
My guess is that being on the losing Stevenson ticket would have hurt JFK somewhat, but not too much. He would have been given even greater exposure across the United States and gotten a jump on dealing wth the religious issue. The Kennedy campaign in 1960 would have spun the 1956 loss to the popular Eisenhower as being inevitable and blamed Stevenson for the ticket's poor showing. The opposition to Kennedy in 1960 was lacking a strong candidate--LBJ and Symington skipped the primaries altogether, leaving JFK to beat up on the underfunded liberal Humphrey; LBJ did not announce his candidacy until the just before the convention; and Stevenson never announced his candidacy in the hopes of being drafted.