Voldemort was a half-blood, so was Snape.
Clearly, pure-bloodness was (probably intentionally, as the whole idea was meant by JKR to parallel Nazi laws) treated the same way as it was treated in Nazi Germany: as was the case with Erhard Milch, when a useful individual was personally approved by top brass, "Wer Jude ist, bestimme ich", to quote Hermann Göring. Heck, they made an exception for Fenrir Greyback despite the whole anti-werewolf thing.
As far as why would the muggleborn want to join? Again, going with JKR parallel, do I need to give you the list of Jews who served Nazis (or profess their deep and abiding love of Mahmoud Ahmadinejad)?
Or someone could simply be a person with no principles. Death Eaters seems to be the winning side - why NOT join if you manage to finagle your way in?