If you witness someone's child doing an avayra do you need to stop him? If he has not reached the age of chinuch – no, but if he is of the age of chinuch, poskim dispute this. The Chayay Odom paskens that for a transgression that is d’oraisa everyone must stop him, but for a d’rabbonon it is the obligation of the father only. (ShA – Rama – MB 7)
Some poskim hold that a mother is also obligated in training her children in mitzvos. (MB 2)
If the child is transgressing for the sake of an adult, he must be stopped in any case. (MB 1)
The age for training of mitzvos-asay depends upon the child's ability and intelligence required for the mitzvah. For example, once he understands the concept of Shabbos he should be trained to hear Kiddush and Havdalla. Once a child has the dexterity to wrap himself, he is obligated in tsitsis. However, chinuch for lo-sasays is from when the child understands when you tell him it is forbidden. If he is too young to understand at all, his father is not required to stop him from issurim. (MB 3)
It is usually forbidden to make a Jewish baby do an avayra or to instruct a Jewish child to do an avayra. You shouldn’t even hand a young child non-kosher food to play with, lest he comes to eat it. (MB 4)
You are not allowed to accustom a child in chillul Shabbos. Do not instruct him to bring you a key through a forbidden zone.(ShA – MB6)
In a case that the child will benefit from doing a d’rabbonon transgression, you need not stop him and you could even give him the issur. For example: washing on Yom Kippur, carrying his siddur to shul without an eruv. (MB3 – BeH “midivray”)
The prohibition of Amira L’akum applies to all issurim in the Torah. (MB 5)
In a case that the child will benefit from an issur (e.g. he is a bit sick) you can instruct a gentile to feed it to him even if the issur is d’oraisa. Therefore you can ask a gentile neighbor to take your baby and feed him chometz on Pesach. The same applies if the child needs gentile wine for his health. (MB5)
When a child becomes the age of mitzvos, it is good for him to accept something as a kappara for avayros that he did as a kotton. If a stolen item is in his possession, he must return it. Even if it is no longer intact, it is good to make any monetary restitution from damages he did as a kotton. (Rama)