Yes, weasels can climb. But you know what, weasels are just VERY narrow and slithery. I doubt there are many weaselproof runs/coops out there. You can spend a buncha money putting 1/2" hardwarecloth over your entire run, including the top (and that IS a whole buncha money) and weasels quite likely can still get in elsewhere, you know?
Personally (and I think many, tho of course not absolutely all, other people feel this way too) there is a limit to how much I am willing to restrict the size of the run (thus the chickens' health and happiness) in an attempt to keep everything out always. Because believe me if I had to cover everything with hardwarecloth, they would only have a *small* run.
I would probably do it if I lived in an area with a known, major weasels-killing-chickens problem, though.
Pat, with one run and the tractor from 1x1 welded mesh, and the others are chainlink with 1/2" chickenwire along the lower couple feet.