There isn't much to do except for what you make of it.
There is the Micky Ear where the little one can play in the water with Swimmer Diapers. There's the Flounder's child care so that you can get away from them for a while. And of course at that age it IS fun chasing them down the asle. Then there's the fun of the sand at the beaches.
We had some of the same toughts and questions you do about taking an infant onboard. So we decided to do a 4 day "test" cruise last October. We were having enough fun that we decided to book a 7-day for this year.
The only thing I can suggest you take your time thinking about is weither you can afford a room with a verandah. If you're travelling with a 1 year old, you may still be facing naps twice a day. That means you will most likely spend a lot of time in your cabin. I know that for us, a typical morning was that we slept late. We got everyone up and out for breakfast. Breakfast took so long (feeding our selves and a one year old starting to want to feed himself) that by the time we got back to the cabin from breakfast, it was time for a morning nap. After the morning nap, it was time for lunch. As by now you might have guessed that by the time lunch was over, it was time for the afternoon nap. So if you enjoy reading, grab a verandah room and spend your nap times reading and enjoying the sea air.