No, they did not.
Southerners in the South where the KKK began supported the KKK because they were the Southerners in the South who created the KKK and filled its ranks. In the South.
Democratics in the North did not support them, Southern Democrats did. Because they were Southern. That means they're from the South. You know who else supported them in the South? Southern Republicans in the South supported the KKK. They even started a faction, the Lily-White Movement Republicans to show how much they supported the KKK and various racist policies of the South.
That's why when Democrats in the South ditch left that stuff behind, Southern Republicans simply got those who left the party to join them because they were already for it. There was no switch. Just people changed party affiliations. The parties didn't switch, the membership did.