In this scenario, Ted Kennedy primaries Carter in 1980 and then wins a very close election(272 to 266) against Reagan. This scenario has already been done before, but let's say the economy recovers faster, the Iran hostage crisis is resolved as cleanly as OTL, and Kennedy secures a nuclear arm agreement with the Soviets. He also pushes through universal healthcare, more anti-poverty programs, and a higher education bill. Lets say he wins comfortably over George H.W. Bush in 1984(the moderate Republicans stage a comeback) and then presides over a successful, albeit quieter second term. The Democrats are still in control of both houses of Congress by 1988, although they probably would have seen significant losses in 1982 and 1986 due to midterm impacts.
The natural question is, who would run for the nomination in 1988? Given the age of the recent Presidential nominees, it is likely that the new nominees would be a set of fresh faces. On the Democratic side, someone younger like Gary Hart or Al Gore would likely get the nomination. The nominee would likely be more liberal than OTL due to the Democratic success under liberal President Kennedy.
Who would the Republican Party nominate? I feel the conservative movement would be broken after 1980, due to Reagan having lost against a party that dumped their own incumbent President and had lost favorability with the public, not to mention losing to Ted Kennedy. And Jack Kemp or any other conservative would likely be considered too extreme for a GOP forced to be more moderate. Would the GOP turn to a "Rockefeller Republican" like John Chafee or a moderate establishment Republican like Howard Baker?