I have taken my daughter with me on several cruises from age 4 months until age three (now). I kind of sucks that you have to pay for them as if they were a third person in the room (on most cruise lines) but that is the biggest downside. A travel agent or web search can help you narrow this down. We like cruisecritic.com.
Of course, with a kid around you can't really party and stay out all night or go swinging from vines in the jungle but you can still enjoy quiet time and beaches and all the rest. My daughter loved beaches and has a shell and rock collection all her own.
I know that Royal Caribbean offer babysitting so you can have a night or nights away, (it costs, but not too much) and I'm sure other lines are the same. RC also offers an hour a day playtime for little kids with fisher price toys that you couldn't pack with you. The staff that works with the kids are trained for it and I have universally liked them.
RC and Carnival offer pack and play cribs for the room or will make up a bed for the baby, your preference. We co-slept and they were cool with that too, even when we had an accident on the sheets and had to get them changed on short notice.
I have found that we get better service from everyone on the ship because they all want to play with the baby. I have never felt that my baby wasn't welcome to join me, even at formal dinners and the captain's reception.
I can tell you that lots of moms on cruises came up to me and said that they really missed their little ones at home, but some said they really needed the time away, too. If you are a burnt out momma it may be worth it to have a few days away.