I know when I was on Carnival you had to be 18 to go in alone. It is probably the same but you never know it may be younger. You can always go in though with like your dad or another 18 year old and you will be fine.
I used Voyagers gym (treadmill) and some free weights during my cruise and had no problem. I had to still stay in shape for track/field season and just to stay in shape. I even used the sauna/steam room with no problem. I was 16 at the time (but in person I look alot older) and was never asked me age or anything. Usually I met my dad there who was already working out, but I saw a couple of kids my age there too so I wouldent worry.
The gym is free and you have to be 16 i think on RC ships
but as long as you seem mature and dont look like your gonna do anything wrong and have come just to work out there not going to say anything . When i was 14 and 15 i always used the gym and nobody ever said anything to me cuz i came to work otu not bother people or mess around or anything so you shouldnt have a problem