I agree with your comment, Ascalepius, right up to the very end.  I've seen nothing that says Republicans support the Klan, but Klan members and Klan-like members support the Republicans.  

It didn't help that Donald Trump was slow in distancing himself from them.  ("I know nothing about David Duke, I know nothing about white supremacy" he said during a tv interview)

Alt-right Republicans have a particularly tricky balance to maintain, between their anti-jewish (KKK) and anti-muslim adherents (comments like yours).