We took our son on his first cruise at 20 months. In retrospect it was not a good idea. We were worn out by the end of the cruise (and yes, my wife bore the brunt of it). But as Penny said, it depends on the kid.
I forgot to mention that when Royal Caribbean rolled out the Royal Tots program they also added an option for you to pre-order diapers, wipes, etc., from the line. These will be waiting in your cabin. That is something well worth the price since it means less luggage.