You can leave your baby, up to 60 days old, with an employee on duty at any hospital, emergency medical services provider or child welfare agency in Texas.
A distressed parent who is unable or unwilling to care for their infant can give up custody of their baby, no questions asked. They must simply bring the infant to a safe haven location and make sure they locate a person to give the child. As long as the child shows no signs of intentional abuse, no name or other information is required. The specific locations and maximum age of the child varies from state to state.
It's safe. It's anonymous. There is no penalty for doing so, however you do give up parental rights to the child. You do not need to tell anyone.