1. An informal "test" to determine the trustworthiness or integrity of a person by judging whether one would trust that person to take care of one's children. I don't trust him to take my car across the country; he certainly wouldn't pass the babysitter test.
2. A test to determine if a household appliance is easy and intuitive to use, i.e., whether a babysitter would be able to use it with ease. The new smart television is radically advanced, but it also passes the babysitter test: anyone in the house can learn to use it in just a few minutes!