For starters, he would be out of an elective office for a minimum of two years. Chances are he would have stood for his old congressional seat in 1954, and probably would have won. Now, when 1956 rolls around and his name is brought up as a possible Stevenson running mate, he might not rule out the second spot so quickly (despite what the old man might have wanted).
Would Kennedy's presence have won the election for Stevenson in '56? Hardly. Ike was popular enough so that defeating him was effectively unthinkable. On the other hand, the '56 election would have given Kennedy a boost in visibility such that gaining a Senate seat in 1958 would have been more likely.
That still wouldn't give Kennedy the necessary clout to gain the top spot on the ticket in 1960, though: in those days freshman senators, especially those with only two years of experience, weren't thought of as presidential timber. That probably means that the Democrats' convention in LA in 1960 would have been something of a floor fight between those wanting a third try for Stevenson, the LBJ forces, and those favoring anyone else (e.g., Kefauver). I'd go out on a limb here and suggest Lyndon Johnson, the consummate used car salesman, would have been able to put together some sort of collection of deals that would have gotten him the nomination with a thoroughbred New Deal-style running mate (Humphrey, same as in OTL).
Now you have Vice-President Richard Nixon and Henry Cabot Lodge for the GOP against senators Johnson and Humphrey. I don't think that Johnson would have evoked quite the visceral support that Kennedy did, especially with the Chicago machine. Thus, I suspect that Nixon would have won a close election (though not quite as close as the 1960 election in OTL).
Barring some cataclysm in office, Nixon would have enjoyed the benefits of incumbency in 1964. Therefore, I would guess that Kennedy would not have challenged him since he would want a situation where a victory would be pretty much assured. Thus...look for Kennedy to be the Democrats' nominee in 1968. The one cloud on the horizon would have been his health, given his bad back and Addison's disease.
The 1968 election would have been a contest between two photogenic, wealthy candidates in their fifties, since I strongly suspect Nelson Rockefeller would have been the GOP nominee, continuing the Dewey/Ike/Nixon tradition (with a more traditional midwesterner--say, Gerald Ford) as his running mate. Kennedy runs with John Connally of Texas as his running mate.
Both candidates have baggage: Rockefeller is divorced and remarried; Kennedy is Catholic, which in 1968 might still not have been trivial. I suspect that Rockefeller would have won in another close election.
By the next real opportunity, 1976, Kennedy is 59 years old, and his health problems are taking a toll. They are sufficiently obvious and sufficiently major such that many Democrats are hesitant/unwilling to throw in with Kennedy, fearing that he would not survive a single term. Thus, Kennedy passes from presidential timber to elder statesman of the Democrats, without ever having occupied the White House. He passes away in 1977 at age 60 from complications of Addison's disease.