The first Klan, in the South just after the Civil War, mostly targeted black people (or white "carpetbaggers" who were trying to help black people). The revived Klan of the post-WWI era was a national movement (strong in Indiana, for example) and targeted Catholics and immigrants as well as blacks.
Since Robert Byrd is from a state with few blacks, he may have joined the KKK more from anti-Catholic bigotry than from anti-black racism.