As i do not have much experience on US based ships i can only comment on what we do here in UK..
When checking in for cruise you are given said paper to sign to confirm all of your party is fit and well, if you cannot sign as someone is or has been unwell in last 48 hours you are taken aside and the staff still check you in, and you board the ship. However, you then have to spend the next couple of days in your cabin, you must spend at 24 hours in confinement AFTER illness has gone, ie if you were ill before boarding you would spend 24 hrs in cabin, If still ill, medical staff will make assessment. Fairly simple in most cases... if serious and they will not let you board, they have to give you written proof as to why and you must cancel at 100% loss, but claim through travel insurance.
Hope this is of some use, we found on our last cruise that most passengers were honest, who would want it on their conciense not to be!