Problem is it's very hard to prove what factors were used in a decision. Person A has a minor criminal record, person B does not? You can just say "B was more qualified" and as long as there's some halfway credible basis for that nothing can really be done. Only if one can demonstrate a clear pattern of behavior might a claim of discrimination go anywhere.

If a conviction is something minor enough that might be expungable, it should be private until that time comes. If the convicted person hasn't met the conditions for expungement, make it part of the public record, otherwise delete all history of it.