I have a friend who will be traveling to Orlando in a few weeks to experience Walt Disney World for the first time with her 3 year old. I know 3 year olds aren't "babies" but parents of toddlers can still take advantage of the Baby Care Centers at all of the theme parks: Magic Kingdom, Animal Kingdom, Epcot and Hollywood Studios.
What are baby care centers? These are special areas in the parks where parents can change and feed their baby(s). There are changing tables, high chairs, a play room, private nursing areas, and a small store selling baby "stuff" such as formula, diapers, clothing, pacis, etc. They even provide some basic necessities for parents of young kids. These areas are air conditioned, which can be nice in the hot, humid, summer months! The play rooms have toys and Disney playing on the TV. There is even a small kitchenette to warm food and bottles and find utensils to use. These centers are great for small children (or parents) that need to take a break during the day or need to have some privacy for feedings. If I'm not mistaken, the first aid stations in each park are located beside or very close to the Baby Care Centers.
The Baby Care Center in the Magic Kingdom is next to The Crystal Palace restaurant on Main Street, U.S.A.
At Epcot the Baby Care Center is inside Future World's Odyssey Center.
The Baby Care Center at Hollywood Studios is in the Guest Services Building inside the Main Entrance.
Animal Kingdom has a Baby Care Center behind Creature Comforts in Discovery Island.
Typhoon Lagoon, Blizzard Beach and Downtown Disney do not have Baby Care Centers, but they are equipped with baby changing stations.