Disney cruises have a nursery for kids under 3. I think you can take kids as young as 8 weeks even. I believe that Royal Caribbean has daycare for young kids as well, but I'm not positive on this.
I took my twins on a cruise when they were 13 months and there was lots they could do. They enjoyed the shows at night, they watched for a bit then slept. They loved the splash areas, which you can do with kids in swim diapers. You can't take kids in the pool with swim diapers, that is correct. They also like the various toddler activities and playing in the kids clubs during open house.
On a cruise it is so much easier to deal with your kids, even when they are young, than it is on a typical vacation since everything is so close together. If they need a nap, you are minutes from your room. Need food, it is free and minutes away as well.