The KKK was originally anti-black and anti-Republican. When resuscitated in the 20th century, it added anti-Catholic to its attributes.
In general, the Democratic Party became the white party by the end of Reconstruction, and so the KKK tended to support it. This should not be understood as meaning that all Democrats supported the KKK; they did not.
The 1910s/’20s KKK was extremely powerful, taking control of state governments in the Deep South, but also having having widespread influence–up to electing governors–in Indiana, Oregon, and elsewhere. Upstate New York was a center of KKK strength too. I’m not aware of any Republican officials who were KKK-affiliated ever.