On the Magic and Wonder they've held a "Playhouse Disney Dance Party" in the mornings hosted by members of the cruise entertainment staff who lead the littlest cruisers in song and dance to some Disney favorites and classics like YMCA and the Chicken Dance. Parents are encouraged to participate as well! They also played Playhouse Disney cartoons on the screens of the Promenade Lounge (where they party was held on the dance floor) during the party. The regular deck parties are fun as well, though you want to be careful in the crowds. For the pirate party, we usually found a spot on an upper deck, so the kids coulds watch the party (when they were old enough to stay up) without fear of being trampled, and so they could see over everyone to what was on the stage itself.
There are some arts and crafts activities that are family friendly, though I found some of them to be difficult when the kids were 2 or under.
The splash area they have on all of the ships is available for use by the diaper wearing set, and is the only water area on the ship that non-potty trained kids in swim diapers can use. My kids loved the area on the existing ship (once they added it), and Nemo's Reef on the Dream/Fantasy seems pretty cool!
On the first night of the cruise, they've usually held an open house in the Nursery and kids clubs. We've taken the kids into both places and played on the equipment and used the time to meet the cast members working there, for when we used the club or nursery later in the week. They hold open house or family times in the Oceaneer Club that are designed for families to come in and play on the equipment in the Club, and you're welcome to take your little one in during those times to play with him on the stuff.
My 23 month old loved the musical shows, but don't know if your 15 month old will. My son loved the characters at a young age, but my DD didn't want anything to do with them until she was 3. There are a ton of photo opportunities with or without characters.
If your cruise stops at
Castaway Cay, there's nothing funner than playing in the sand! They have wagons and beach strollers available for use on a first come, first served basis, that are great for toting kids and stuff around the island. We generally find a shady spot on the beach away from the water, and let the kids dig in the sand. You can buy beach toys on the ship or island, bring them from home, or order some in a gift basket. You may also be able to borrow some from the kids club on the island, since many folks leave their toys behind after their cruise rather than lug them home. Castaway Cay now has Spring A Leak, which is a fresh water splash and fountain area aimed at toddlers on the island that opened in 2010. We've also rented bikes with the baby seat on the back and biked around the flat bike paths for the afternoon.
