Does the hand become impure by touching a child's trousers that are wet with a moisture of doubtful nature, i.e., is it water or urine?

A thing is not deemed impure merely by doubt. The original state of a child's trousers is purity, even if you doubt it. You are not required to wash your hand or anything else as long as you do not know for certain that it is impure; not all moisture is impure. The default state of things is purity, and they are only judged as impure by a confirmed matter. There is no need to investigate the source of the wetness.