No fur value. You can sell one occasionally on taxidermy.net but not for much. They are hardly worth fooling with. They do seem to make good bait for bobcat if you catch one.
As to method the standard way used to be 110s set in their tunnels. Very efficent and very humane but illegal now. Most mountain beavers in WA are trapped for damage control so permits are obtained to set padded jaw traps,#1 or #1 1/2. Not so humane but trappers are kind of forced to it. If you get a permit be sure and stake with no slack as they will take the trap down deep and get wound up making it hard to retrieve them. They are somewhat colonial so there will usually be several using a series of tunnels.
Eating them
They do not smell good. Survival food only.
On another note though if you find a colony of them, that is a heck of a good location to trap bobcat.
Besides that everything uses their tunnels, Mink, weasel and look out
civet catsBesides mountain beaver a lot of small rodents and believe it or not birds use the tunnels especially during nasty weather. It makes it a predator draw.
