Firstly there is no need to have your active monitors plugged into a UPS. You are just adding extra load onto your UPS and it is unnecessary. Your UPS really only needs to power your computer and monitor so you can do a save in the event of a power failure.
Secondly, yes powered or active monitors do draw power even when they are doing nothing. We call this the quiescent power draw or idle power draw. Over a year it would certainly add up to your power bill.
You really should shut everything down when you are not using your studio. It is NOT a good idea to leave anything running, especially if you go out for any period of time. People who leave things on permanently are simply wrong and silly IMO. There is nothing to be gained from it. I have been turning everything off for the last 32 years and have never had anything fail as a result. Windows also likes being shut down and rebooted as well.
You are setting yourself up for a fire hazard if you do leave things on permanently. What happens if something decides to set itself alight while you are away. It happens, houses burn down! Put your speakers on a switchable power board and simply flick one switch to bring them on and off. And in the right order of course. Speakers last when powering up and off first when powering down.