A babysitting job isn't very demanding, or we wouldn't have so many teenagers doing it. If someone seems to be qualified, then they probably are. If you're okay with hiring a murderer in general, I don't think they're notably less likely to babysit correctly. Let me know if I'm wrong here? (If you're not okay with hiring a murderer in general, then ignore this comment.)
For a sex offender, it depends on what kind. For example, a former prostitute is no less qualified than anyone else. But sure, some kinds are relevant, and should disqualify people from a narrow slice of jobs.
The problem is when people have supposedly served their time but can't get 95% of jobs.