Off.  For multiple reasons.  IF we are lucky enough to get a truck to it, leave it on until back at camp on the meat pole.  But if you are quartering and packing, off immediately.  The hide holds in a lot of heat.  If you want tasty wildy game, start it cooling down ASAP.  Also, leaving the hide on will allow hair to get on the meat.  a few wont hurt but a lot of hair will transfer gamy taste to the meat.  Skin the hide and get it into a game bag.  If you skin/cut the glands or the pelvic area out(which you should)  use a different knife or clean it before cutting meat   Take care of the meat in all ways from the beginning and you will appreciate it on the dinner table.  
I know some folks leave the hide on the quarters in order to keep some of the meat clean. In my opinion, a good game bag is a better idea.  And not one of the ones you get for a dollar at walmart. a good cotton one or a sheet is really good.  The new material bags are light but i don't think they breath quite as well.
 
I am always amazed by how many people say "it's how you take care of the meat", and then they don't.  I see folks driving back home from a hunt with the hide still on,.  Im sure they give packages to their friends who cant understand why you would hunt when the meat tastes so nasty  
thats my 2 cents anyway