I would have to say it was probably during the late 70's into the 80's. The Republicans didn't come around until Lincoln and the Civil War, at which point they were an ultra-liberal party from the eyes of the solid-south. Even into the 60's and civil rights, the democrats were still predominant in most of the south and opposing integration (wasn't it Arkansas where the National Guard stepped in?). Kennedy wasn't even that big on civil rights, but LBJ was able to use his death to push it through. Texas has been Republican for a very short time, electing Bush in '88 was a change.

I would probably mark the change with Carter, who gave away the canal and let Castro ship all of his criminals and insane into Florida.