Iris:
I'm the family cruise editor here. Disney is the only line that has a fleetwide nursery for infants and toddlers, which you and the toddler's parents might make use of. Carnival has limited hours when it accepts toddlers into its program for group baby sitting. Royal Caribbean won't accept toddlers into its youth program but does have private babysitting for children over 1 year old. We brought our 7 month old son on Disney last year and felt very comfortable having him stay in their nursery. (There is an hourly fee and you have to make reservations for him/her to stay there once aboard ship.)
Let me know what you decide.
Smooth sailing,
Luisa Frey GAynor