Best way to trap a beaver is;
1) Get a proper sized trap
2) go out during the day and tear a hole in the dam, enough to get the water running through it, audibly.
3) set the trap, under water, in front of that hole.
or just set it in front of the den, under water. Baiting the trap with a small piece of fresh cut aspen works too.
The tails are good eating. The pelts don't get much in the way of money anymore, and this time of year they wouldn't be great anyways... but fleshing and stretching them isn't too much work, and beaver fur is incredibly durable, makes for great trim on moccasins and wrap-arounds. Would make good string silencers and quivers, too.
remove the castors and freeze them, they're worth money.
Properly butchered and prepared, the meat is pretty good. But it's easy to make it awful, too. Makes good dog food, as well, sled dog racers love them. Some guys use them for bear baiting... but that seems wasteful and wrong in about a dozen different ways, for me.
I've got beavers behind the house, like New York has rats. My local R.M. has a bounty on the tails.
if you do decide to shoot them... a spring-fork made from a green sapling can work to recover them when they sink. Split the trunk for about three feet, open it with a stick wedged in it, and use the "spring" tension to grab the beaver under water... basically... "spear" the beaver, knocking the wedge out, and the sapling springs shut, giving you enough grip to pull it up.
