I'm not too worried about someone stealing the trailer off my bike, but if I were to store the trailer outside, I'd definitely lock it to something. Most likely I would get a tiny cable and secure it with a mini padlock I already have. Using something like that around the chainstay to lock the trailer to the bike sounds brilliant!
There might be a way to just use the padlock with the coupler itself instead of the cotter pin thingy to secure the trailer to your bike, but it would have to have a really long shank, and mine doesn't.