From my understanding, if a sailor or officer was violently seasick and didn't get over it after a day or two, that's a liability to the ship and was bad for morale, so it's in everybody's best interest to transfer him to a shore role (which probably limited their chances for promotion and so some refused). There were a ton of Navy jobs that didn't involve being on a ship. Some guys probably lied about how sick they were or just hung on out of sheer stubbornness, and they made it work, but that depended on their NCOs and CO letting things slide.

But, generally you'd want to figure this out before the guy was deployed somewhere where him being half-dead from vomiting the past 10 days straight might or not might matter. So, everyone went on a training cruise during basic training or OCS at least in part to see how they handled being on a ship in deep ocean. Worst case,l scenario, you find out on your first Atlantic crossing or the trip to Pearl Harbor, you suck it up and do your best until you get there, and they reassign you there.