What about Norovirus you may ask? Noroviruses are a very common group of viruses (only the cold virus is more common) that can cause stomach and intestinal illnesses. The virus is spread from person to person quite easily and is often brought on board a cruise ship by a passenger or crew member. It can also be spread through contaminated food or water. There seems to be a report of Norovirus outbreaks on cruise ships with alarming regularity but this is because it is so closely tracked on board with each ship having to report instances of illness (including zero illnesses) for each itinerary they sail. These results are then made available to the public here. The Center For Disease Control provides suggestions about preventing Norovirus here. There also appears to be no correlation between inspection scores and instances of Norovirus outbreaks so it is important to me diligent about sanitation regardless of how well the ship you are travelling on scores.
Washing your hands regularly, carrying and using hand sanitizes, not touching public parts of the ship are always ways to limit your chances of getting ill. One area we really like to avoid touching are handrails in the stairways. We will lean our hip or forearm into the railing to maintain balance and avoid touching a part of the ship that is regularly handled by large groups of people.
Jan and I also only drink bottled or boiled water on the ship.....or ice cubes sanitized in various types of alcohol :-). As we are sensitive to changes in water (often laid in at each port of call), we find the consistency of bottled water good for us and it eliminates the possibility of drinking contaminated water.
We have been very lucky. On fifteen plus cruises we have never come down with anything resembling Norovirus. We have, on longer cruises, developed a dry cough. This kind of "cruise cough" is often attributed to the air conditioning system in stateroom dying out the air too much in the evening. I have only noticed it on cruises of ten days or more, and a couple of cough drops seems to help out if this should develop.
Don't worry about illnesses at sea. Take regular precautions and you will find yourself fit as a fiddle upon your return to dry land.