Start with Trans Union, they have a child specific application so you can find out if your child's SSN has been used by identity thieves: http://www.transunion.com/corporate/personal/fraudIdentityTh... If they don't have any reports, there's a good chance you're probably ok.

You can also apply to put a security freeze on your child's SSN. State by state laws and application process here: http://consumersunion.org/research/security-freeze/

And then there's the myriad of companies who can give you protection for a monthly fee:

AllClearID: https://www.allclearid.com/

LifeLock Junior: http://www.safety4yourkids.com

Also, Experian has a monitoring service as well specifically for kids: http://www.familysecure.com/

Hope this helps.