We did a Canada/New England cruise when our daughter was one. She did fine, even in the dining room. It's true that your night life will suffer, so if that is an issue you might want to wait until they are older. They do have babysitting in the evenings if you do want a few hours away, for a fee. They also have time set aside during the day when you can take young toddlers into the play room. I believe it was when the room was closed for lunch on sea days - you can't leave them there of course, but you can stay and play for an hour or so.
One note, the shop on the ship did not carry diapers, so make sure to bring enough with you or make plans to pick some up in port. My husband and son did a fishing trip, so I put my daughter in the stroller and we did a walking tour of Halifax, complete with a stop for diapers at CVS.