According to the accounts in The Crosswinds of Freedom (James MacGregor Burns), the Democrats weren't especially united after the 1960 convention in Los Angeles. In particular, Adlai Stevenson's backers, led by Eleanor Roosevelt, were especially disappointed and lukewarm at best to the Kennedy candidacy. To mend fences, Kennedy visited Eleanor Roosevelt at Hyde Park shortly after the convention and by all accounts won her over--and with that, the support of the Stevenson faction.
Now: what if that visit went sour? Let's say it was something simple: maybe, like Nixon IOTL, Kennedy banged his knee or something on a car door, or tripped / fell / tore his suit pants--anything that would put him off, leading to at best a half-hearted rapport with Mrs. Roosevelt. And that in turn means he gets weak support-if any-from the Stevenson faction: sort of "well, at least he isn't Nixon, but..." feelings.
What happens then in the election, with diminished support from the coastal members of the party (recall that a lot of Stevenson's support came from the northeast as well as the Pacific Coast states)?