There's nothing that makes murder specifically relevant to babysitting. If the implication is they wouldn't be hired, that implication applies to every job. So they'd be unhireable. Which in practice is more or less the same as "they should still be in prison".
Should this person not be allowed to work with children?
> If the implication is they wouldn't be hired, that implication applies to every job.
Unhirable for one, high trust job doesn't mean unhirable for every job. For example, I don't care if a sex offender washes my dishes at a restaurant, but I wouldn't want them teaching my preschoolers.
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.