Depends on what they've done to be honest. Did they kill a man or rob a bank? Because both a serious but mean completely different things morally. Plus when looking at the crime (or any action to be honest) you have to look to WHY they did it. If they could justify it then I could probably understand, although some things would be unexcusable no matter who they are (i.e sexual abuse crimes).
The only problem I would have would be on the things that are in a shade of grey, like violence, but if I knew a person well enough you would be able to understand their psyche about the whole thing and hopefully talk about the incident/situation that happened.
For the most part I'd probably be ok with most things and it wouldn't effect the relationship, as some of my close friends have had bad situations in the past, but are not bad people but in fact quite the opposite.