Legal Question in Family Law in Pennsylvania

I'm afraid that there is not enough information here to offer you a truly meaningful response. A "convicted criminal" could be someone who was drunk and disorderly outside of a bar five years ago, a registered sexual predator, or anything in between. You do not mention that this person has conducted himself in a way to create concern in a reasonable person. I do not know how old the children are, how often they are alone with him and why the mother needs him to babysit, or whether he resides with the mother. I know nothing about the environment in mother's home, why you think it is a bad one, or what other parenting issues the mother may have to lead you to think that she has impaired judgment where the children are concerned. All of that said, your husband had better have something far, far stronger than "Mom's boyfriend was once convicted of a crime!" before a court will forbid the children from having a relationship with their mother. Be very glad of that... it means that the mother can't do it to your husband, either.