We just travelled the Dempster & we pulled our 15' trailer (and our friends had their 17') up the Dempster as far as Inuvik. We left the trailers there & drove the rest of the way trailer-less to Tuktoyaktuk for the day. While the Dempster is a rough ride (& both couples got one flat each), it's definitely doable with a smallish trailer.
If you still want to leave your trailer in Dawson City though, you can't leave it at a Yukon government campground, as they don't allow unoccupied campsites for more than 24 hrs. So, you'd have to leave it at a private campground (i.e., Dawson City RV Park, Goldrush Campground or Bonanza Gold RV Park).