Byrd has been a friend to the progressives, for a good many years. He's the man who stopped the Clinton impeachment. He's called Bush to account more than once. When he was young, and stupid, he belonged to the KKK, probably for the same reason Harry Truman joined, so that he could get votes. It is a pretty safe bet that Byrd was, likewise a Mason, a Shriner, a Moose, a Kiwanian, an Elk, an Eagle, and anything else that would take him. I wouldn't be surprised if he had some affiliation with the Knights of Columbus. Politicians do all kinds of things to get themeselves elected.
Joining the Klan was dumb. Though I'll bet it seemed like an excellent idea at the time.