As I recall, Robert Byrd wasn't just in the Klan. He was a person of authority within the organization.
As I also recall, he has never publicly separated/condemned himself from this organization. Which would not make it ok even if he had.
It has often been said that Byrd was a Chief Wizard. I don't know if that's true or not. But considering his political prominence, it's likely enough.
There are numerous statements out there from Dems and figures in the media that David Duke disqualified himself from public office for life because he once belong to the Klan.
But the rules for Dems and Republicans are different.