Give out baby's social security #?

My mother-in-law wants my daughter's social security # so she can open a bank account for her. Everyone has been asking for the # and I've been telling everyone that I'm not comfortable with giving it out. I haven't even used it. I just opened another bank account in my name and put her as the beneficiary. Am I being too cautious? I just figure in this day and age, some things should be protected. Especially if there are other ways to get around it (FYI she got pi**ed off and hung up on me. Any ideas how to handle that?)