The simplest way to do this, IMHO, is to butterfly Nixon's Southern Strategy. Without Southern conservatives in the Republicans, the balance of power in the party will not shift to the right, and the conservative Western/Midwestern wing will be balanced by the Moderate/Liberal Northeastern/Northwestern wing.
So there are two possible scenarios. One is the Dixiecrats never leave the Democrats, which maintain a right-wing element until today. This probably means civil rights are not pushed under LBJ as they were IOTL. Perhaps they are delayed an election cycle or two and passed under a moderate Republican? Probably leads to a much uglier U.S., quite honestly.
The other option is the Dixiecrats do break, but they walk off and form their own independent party. The positive of this is civil rights could be passed as they were IOTL. As IOTL, this suggests that an informal "conservative coalition" will remain in place between the Dixiecrats and conservative Republicans, so legislatively it may not be all that different from OTL.