Not all southerners did, but those in the Dem party didActually, Southerners fought ending Jim Crow tooth and nail. Of course, back then, they were all registered Democrats, who later deserted the party for the GOP.
and since white resentment and bigotry are core themes of GOP politics today, and Trump openly plays to white bigots, the dishonest insinuation that the Southern Democrats of old were liberals is obviously false.
But it is a trope that originated on right wing talk radio, which has always pandered to the white resentment element.