You'd have to start before Truman, since he was aggressive, at the time, pushing some civil rights legislation and that led to the beginning cracks of the Solid South.
Maybe Truman refuses to run and the conservative Democrats are successful in getting John H. Bankhead II on the ticket with FDR. Bankhead was from Alabama and was a staunch opponent of civil rights. Bankhead ascends to the presidency after Roosevelt's death and runs a far more conservative administration domestically than Truman.
This allows the Solid South to continue supporting the Democrats in '48, as Truman lost Alabama, Louisiana, Mississippi and South Carolina. I think Bankhead wins 'em all rather easily, though might not do nearly as well in the west as Truman. Either way, even if Bankhead loses, it radically changes the dynamics of the next four years. If Dewey wins, he certainly would have been more progressive on civil rights than Bankhead and that would have only continued to alienate the southern faction of the U.S. from the Republican Party.
In four years, with a depleted economy, and a dragging war, Dewey probably loses. So, do the Democrats continue with Adlai Stevenson or do they try to shore up their southern base by nominating Estes Kefauver or Richard Russell, Jr.? It'd be tricky, because northern liberals would certainly still hold a great deal of influence at the convention. In fact, in reality, Richard Russell, who gained early traction, was opposed by liberals from the north because of support of segregation.
But maybe, after Bankhead's run four years prior, the liberals in the party are squeezed out, and or support the more liberal Dewey. Either way, in this scenario, let's say the Democrats do the unthinkable and nominate Russell. If he goes on to win the presidency, the entire dynamic of the U.S. shifts. Not only do you lack a pro-civil rights president at the infancy of the movement (Eisenhower, in reality, was far more progressive here than Russell would be), you'd have quickly alienated a great deal of blacks who were brought into the party during the Roosevelt administration.
At that point, you can pretty much alter whatever from this point forward. Would Russell win reelection? If the economy is booming, it's possible, even if he's starting to really get blow back for his support of segregation. If he does win reelection, that sets up an interesting election in 1960. Whereas in the OL, Kennedy ran, in this TL, he probably wouldn't be viable with a growing coalition of support in the south for the party.
At that point, the Republicans, who had already made gains in the south by this time (Kennedy struggled in the south) would probably lean more toward the liberal north east for their support. Who knows who they put up to run against the Democrat (and who the Democrats would run...maybe Harry Byrd?), but either way, the Moderate Republican probably wins and becomes president in the 1960s.
At this point, Moderate Republican pushes through a more progressive civil rights agenda with the help of liberal Democrats in the north and at the opposition of conservative Democrats in the south. This only helps widen the gap between Republicans & Democrats in the south and sooner or later, a great deal of the liberal Democrats gravitate over toward the socially and economically progressive Republicans.
In '64, the Moderate Republican wins reelection, continues supporting civil rights, while the Democrats struggle to field a candidate acceptable to the entire party, especially the south, which finds itself politically pinched because of some of its regressive views.
Eventually, though, in '68, the Democrats run a conservative whose rhetoric isn't nearly as sharp on race relations. Possibly George Smathers runs, who was an opponent of civil rights, but did vote for the Voting Right Act (which would have probably been pushed by Moderate Republican instead of LBJ). This makes him acceptable to the southern Democrats and moderate Democrats alike and he goes on to win the presidency.
Whomever the Republicans ran carries a great deal of the black vote, does extremely well in the northeast, but loses the entire south and pockets of the Midwest. The Democrats have built a solid firewall through Missouri on down to Texas and from Oklahoma through to the Atlantic Ocean.
From there, you've got the foundation for a very conservative, populist party that is dominated by southern evangelicals and other religious folks who would almost certainly oppose abortion. By the 80s, the Democrats are seen as the conservative party and the Republicans the more moderate-liberal party - both domestically and internationally.
Even so, the Democrats are good at keeping with FDR's populist message and still keep a good deal of his economic rhetoric. The party is seen as that of mostly working class, white men and southern. The Republicans are a party of white women, liberal elites, intellectuals, blacks and middle class folk. Their region of dominance is the northeast.
Pat Robertson, in this reality, is a Democrat. So was Jerry Falwell.