H.,
We did the Disney Cruise last summer. My children were 4 and 7. It is AWESEOME! Unlike other cruise lines, it is completely geared towards families. Lots of plexiglass so no one falls over, characters everywhere, shows the kids loved, very kid friend food. Disney's private island is amazing. Captain Jack's boat is sitting right in the bay, what kid wouldn't love that?
I frankly think you could take a baby and it would be fine. There were lots of babies cruising with us. Their child care on board is fantastic, my kids LOVED it and wanted more of it. And if you don't want to use the childcare, there is plenty to do and take your kids too. Disney is very careful about cleanliness on board, hand gel and wipes are everywhere. The cabins are larger than other cruise lines and family friendly.
Obviously, I highly recommend it and would love to go again. If you go, book a massage for yourself on the private adult beach at Castaway Cay, so worth it. I had my husband watch the kids while I went and it was the best money I ever spent. :) H. this helped you a tad.
One side note: I love disney. We did take our kids to Disney world when they were 2 and 5. Some would say it was young. They loved it. They were young enough to truly still believe in the magic. My daughter thought she really met all the princesses and that they were real. Too cute. But of course I'll take them again. I think Disney is something to enjoy throughout life. Good luck!