The Civil Rights Act still passes with the increasing pressure from and visibility of the Civil Rights movement post-Birmingham Campaign, just more narrowly. Most of what we know as the Great Society programs are likewise passed by congress as a New Frontier-branded legislative package, albeit trimmed down more to middle class-oriented consumer and environmental protections. Minor note, but since I recently did an article about it, there might not be the unprecedented federal investment in education that there was IOTL since Johnson was influenced by his time as the head of the Texas National Youth Administration. Goldwater still gets the nomination on the backlash to all that, but Mississippi Governor Ross Barnett had a particular hatred of Kennedy and IMO would have run third-party- he was more of an arch-segregationist at core than Wallace, who probably just ran in the primaries as a grift.
Kennedy probably doesn't have some major scandal yet like some people think- a year probably isn't enough time for an infidelity scandal to break, and the White House physicians stopped Max Jacobson's meth treatments, so no public psychotic breaks like the alleged Carlyle Hotel incident could have become- nor is Johnson swapped out like some people speculate. Kennedy still commits to direct involvement in Vietnam for the same reasons as Johnson, but there would definitely be some friction with the CIA on Cuba, as Kennedy thought he could engineer some kind of Cuban Thaw. He was nervous about how it would be perceived, but was aiming for it nevertheless and receptive to Castro's overtures.
Overall, minus Barnett, Goldwater does a little better without Kennedy made a martyr, flipping Idaho and overcoming the backlash to his opposition to farm subsidies in Nebraska at least. But he and Barnett neutralize each other in the non-Deep South, and he was a little too extreme for the time to do much better.
President John F. Kennedy (D-MA) / Vice President Lyndon Johnson (D-TX) ✓
Senator Barry Goldwater (R-AZ) / Congressman William Miller (R-NY)
Fmr. Governor Ross Barnett (I-MS) / Congressman Bob Sikes (I-FL)
I remember the BBC having a take on that alternative history:
Fun episode, but major Red Dwarf fans like myself will note that the show is a comedy.