The phrase "to keep an eye on one's baby" means to watch and take care of your child or loved one, making sure they are safe and out of harm's way. Full definition
Keep in mind that baby tubs are not safety devices, so you should always keep an eye on your baby while he or she is bathing. (pregnancymagazine.com)
There are some general guidelines parents can look at to help keep an eye on their baby's progress. (americanpregnancy.org)
While they can be a great way to keep an eye on your baby when she is sleeping in her nursery, they can have some significant problems, including poor range and interference. (verywellfamily.com)