A recent NY Times (http://www.nytimes.com/2015/04/18/your-money/a-childs-vulnerability-to-identity-theft.html?ref=your-money) article talks about how thieves are getting access to Social Security and other personal information from health insurance company Anthem. A 2011 study of 40,000 children caught in a data breach found that some one else was using 10.2% of the SS numbers.
Thieves use children SS numbers because they have a clean credit report, making it attractive to someone else who wants to start their credit history. Also, kids do not typically check their credit reports.
One way to protect your kid's identity is to put a freeze on their credit reports. Equifax, Experian and TransUnion are the three agencies that track your financial history and sell this information to companies we want to do business with. So, you put a freeze on credit reports with these agencies.
Credit freeze is a better option than fraud alerts as credit freeze is more stringent. But these agencies are not supporters of credit freezes, as they lead to more administrative work. Every agency has it's own procedure for setting a freeze, so check out their websites.
The credit freeze however won't stop all identity thefts, as some thefts do not need credit reports. For example, a SS number can be used to file taxes and get refunds, to gain legal authorization to work, and to enjoy health care benefits.
So use some additional precautions to avoid identity theft. Do not carry your kids' SS card everywhere. Always question people that require their SS number. Also, warn your kids to be careful when online and never give out their SS without your permission.