Hi, Maura! My four children have been visiting WDW since they were infants, and "babies" are considered
no cost at WDW until they reach three years old. So, I personally think that bringing a child between one and three is a great time to enjoy them in the parks. Before they are one, you can still enjoy them, but the difficulty in bringing them (frequent feedings and naps, diaper changes, long hour days, etc.) sometimes outweighs the fun.
With that said, there's a broad definition of "babies" when you consider that you may be talking about a newborn up through the age of walking and talking. For the sake of this answer, I'll refer to mainly toddlers up through preschoolers (almost 1-3 years). There are many things that little ones can enjoy at each of the four parks including all sorts of attractions and entertainment. I recently answered a
question for a guest that wanted a list of things that her 18-month-old child could see and do at WDW. (Click on the link above and maybe that list will also help you plan.) Depending on your child, meeting the characters and stopping for impromptu street entertainment are great ways to spend a day in the parks with little ones. Then, there's great dining that always caters to even the smallest guests.
Lastly, you'll find baby care centers at each of the parks which are great ways to take a break from the crowds and/or the weather to regroup for your baby. They are usually located near the First Aid stations and are easily identifiable on park maps. Remember to bring your own stroller and supplies, and pack an extra measure of patience. Let me know if I can help you further. Have a wonderful day and a magical vacation!