Lets face it, kids get sick often. Think about your normal life at home and how often you come across a sudden cold, sniffles, belly aches and the like. Take this into consideration when packing. Make sure you have all the normal medications such as cough syrup, aspirin, heating pad, eardrops, etc. before you leave. Many of these items will be available onboard, but it is worth taking your own to be sure you have what you need, the type your child likes, and that you don't get cheated on the price.
If you have one, it is worthwhile to bring a collapsible stroller with you. These are usually available for a fee onboard but again, its worthwhile to have your own. The ships are very large and there will be a fair amount of walking around to get from one place of interest to the next. Just going to dinner or getting a snack will usually involve a few elevator rides and a stroll down a long hallway or deck path. Be prepared here, you want the stroller you and your child are used to using.
Many cruises offer some sort of baby sitting or child activities where the parents can get away and have some time to themselves. Don't count on this, however, and be sure to check in advance. Many cruise activities are scheduled throughout the day but there will certainly be 'down time' at day or night when your children will need to be entertained. The best advice is to bring along some of the same activities you have at home. Books, games, video games all work well when there is time to be spent in the cabin. You may think "hey, I'll be on vacation and not spending my time in the cabin"...but you will, trust me.
In conclusion, the moral of the story about bringing your kids on a cruise vacation is to bring the comforts of home with you. Anything you can pack that is not too overbearing to carry you should consider bringing. Following this advice will make for a much more comfortable and relaxing vacation for you and your kids. Packing to stay healthy on a Cruise Vacation