NO ONE supports the KKK. The Democrats used to but that is far as it goes. The Republicans never supported the KKK at any point in their existence. The South stopped supporting the KKK about the time the South dumped the Democrat party. The stigma that the South is full of racists needs to end because that is no longer true. Just because the KKK uses the Confederate Battle Flag does not mean the South supports them or even appreciates their use of that flag.
I don't know if all Klansmen are Republican or not, nor do I care. I care nothing for the KKK. If they are all conservative then that is their own business and the clearly do not understand freedom, just like Democrats.
The Democrats continually are trying to push the KKK and the Neo Nazis off on us mainly because they want to disassociate themselves from the times when the supported both groups.
In a way Democrats still sort of support Neo-Nazis. Nazis are Socialists and the Democrat party is the Socialist party. It makes no difference if they want to call themselves Democratic Socialists (what Bernie Sanders and Vladimir Lenin called themselves), National Socialists (what Hitler called himself), International Socialists (what Josef Stalin called himself) or Fabian Socialists (what a bunch of British twits called themselves) the horrors of Socialism is fundamental to all Socialist ideologies.