I mean, it depends really. 1960 will probably still be Kennedy/Nixon, and yeah, Nixon's running as a moderate so that might be to his disadvantage, but Kennedy is inexorably associated with the Democrats. Now obviously Johnson isn't going to be on the ticket (most likely he'll choose a Northern senator like Scoop Jackson ITTL), but seeing as a load of Southern Democrats (and quite possibly some Northern Democrats) are bound to go down in the KKK purge, the Northern electorate might be dissuaded for voting for a Democrat because they're afraid Kennedy'll be a Klansman in disguise.
Additionally, the downfall of the KKK wouldn't eliminate institutional racism entirely. The Southern electorate might be temporarily embarrassed and vilified by the North, but they'd still be racist as fuck - they just can't win on explicit racism. That whole Lee Atwater quote comes to mind:
So there are two options. The first is that Kennedy wins as IOTL, and in 1964 Goldwater downplays the opposition to civil rights and massively plays up the neoliberalism. Efficiency in the private sector, aspirational politics, Hayek and Mises and Friedman, basically the tax cut bit onwards but sixteen years earlier. There's a very good chance that Goldwater defeats Jackson or whoever ITTL, and oh Lord 1968 would be something then.
If Nixon wins, the Democrats might try and reorganise as more of a Christian democratic party after the German CDU. This way they'd (a) rid themselves of any racist associations, keeping Northern conservatives on their side (b) keep the evangelicals aligned to the Democrats, thus keeping Southern conservatives on their side (c) probably not turn neoliberal after 1980, assuming they adopted the CDU's ordoliberal social-market economic policies. This way, Reagan stays a Democrat, so you might see something like a Reagan/Connally ticket in 1968 or 1972.