Ultimately you need to go back to 1912 and keep Roosevelt and the progressive wing from splitting the party- it was his split that allowed the conservative wing to take control of the Republicans, giving them carte blanch to shape the party how they saw fit. They were conservative prior to the New Deal, with the three Republicans Presidents prior to the New Deal, Harding, Coolidge, and Hoover, notably conservative (Hoover is debatable, having begun as a progressive and turned to conservatism).
So really you have to start there, not in the New Deal or Civil Rights era because by then the wheels were already in motion and the direction of the party was as clear as day, but instead for the Republicans to keep 'liberal', you really need the progressives to win out over the conservatives in that crucial period in the early 20th Century.