I just finished workig on ships for the last two years and yes passengers and crew member get together all the time. If caught, it will get you fired immediatly. However, that all depend on how it is and what their position is. I once had a captain that would make out with a different passenger every week, and do so in front of everyone. He didn't care, but he was the boss. There was no one on the ship above him and therefore no one to answer to.
It happens from the captain all the way down the lowest man on the crew. You just have to be discreet about it. I once had a passenger kiss me in on of the bars and out in full view of everyone. I did get written up, but that was it. Again it has a lot to do with your position. If that were a waiter or houskeeper, they would have been fired on the spot. It's not fair, but it's the way that it works.