Ah, but this isn't an " open or shut " case
---Some say yes, leave it open--some say no, close it. Some say they like to sleep with the balcony open to hear the waves, water, etc. It does , as far as I know put a load on the a/c system,and as well, will fill your cabin with damp, humid air and then the next day you have to run the a/c wide open to try to cool off the cabin again.And too, as was mentioned, no matter how many times it's announced not to do it, there's always some nut who will throw cigarette butts overboard and they can and do blow back into the balconies at times.
One could very easily blow back far enough to get the curtains and bedding on fire. One fine morning I went out onto our balcony and a cigarette butt had landed in the white plastic chair and had burned a place a couple inches long into the plastic. The chair was usable but for all practical purposes was ruined. And so, some are against leaving it open.
One thing you need to be careful of and especially so if you have a small child or children is to remember that if the balcony door is open, day or night, and you open the cabin door, there's usually a very strong surge of wind that will slam your cabin door especially hard and if you should have a small child in the way, it could get hurt very badly by the force of the slamming door.
Having said that, I sleep with mine shut as I don't want to get a cigarette butt into the bed with my butt and don't want to have all the humid air pouring into the cabin and then have to spend the next day with the a/c full tilt to try to cool it off, so I can open the door again that night an start all over again the next day.
So. open or close--Tis your choice--is it better to try to hear a faint splashing now and then and risk being caught up in a fire, have a humid , damp cabin, or tis it better to comply and shut the door, enjoy the motion of the ship and sleep in comfort!!
And besides, there could be cat burglars climbing about on the balcony rails, looking for that open door through which to enter and make away with your valuables while you are lying there snoring away!