A easiest way your child's identity can be stolen is if someone gets a hold of their SSN. One way a criminal can obtain this is by you accidentally signing up accounts for fraudulent businesses using your child's social security number.
A second way your child's information could be stolen is through social media. Criminals can get your personal information by piecing together things that they put on social media,
A third way your child's personal info could get stolen is if they get their hands on your childs wallet. Their is a lot of info they could have in their wallet which could lead to their identity getting stolen.
Video on how your child's identity could get stolen
You can prevent your child's identity from getting stolen better by looking into and limiting the companies you give their social security numbers to, have them put restrictions on whom can see their information on their social media accounts, and keep all their passwords secret without sharing them to others.