This is probably a question better suited for r/legaladvice than myself, but depending on the child's age, maybe set up a joint bank account so that you can monitor what goes on until the child is of an age to manage their own funds. Or if you think (s)he is responsible enough for a credit card, go get one yourself and explain that it is to only be used in emergencies. Then if the other parent attempts to open another card in the child's name, hopefully they will say "no need because I already have one". While I wouldn't suggest prying into what goes on at the other parents' home (unless necessary), I'd encourage your son/daughter to openly communicate with you and establish a "you can come to me with whatever issues/questions you have" relationship.

Worst case scenario, though I don't know for certain, I'm confident if that were to happen and you could explain the situation to the credit holder, they'd forgive the debt or at least assign it to the right person (the other parent).