I don't think it's your place. If your boyfriend is around it's his responsibility. When it comes down to it, if he leaves you to watch his child I think you ought to tell him you aren't going to watch the child. Who watches the child is between your boyfriend, the child's mother, and maybe (if they need a babysitter) one the child's grandparents.
Having said that, and assuming you're going to be around this child, young children behave well for adults who are kind, friendly, and respectful but who have a way of letting the child kind of take it for granted that the adult is the "leader" when they're together. With very young children it pretty much takes constant, patient, supervision and calmly letting the child know if he heads for something he shouldn't be getting into.
If your boyfriend's child is older the chances are good that he will "rightfully" feel you have no right to "discipline" him, so he may act accordingly. If your boyfriend cares about, and respects, his child he won't leave him in an unhealthy situation. If he cares about you, and the child is older, he won't expect you to be responsible for an older kid who knows you have no business trying to "discipline" him.
This is going to sound awful, but a lot of fathers (especially young ones, but even older ones) will try to get custody of their children, or will have their visiting time with their children, and try to foist them off on some woman because - when all is said and done - they either don't know how to effectively parent or else have to work. That kind of situation isn't good for the child, and it isn't good for any woman the father expects to pick up the slack when it comes to his own child.