I have had a lot of experience with beaver incursions and have both trapped and shot them. My 3 acre pond is just upstream of
a large reservoir (8000 acre) with an estimated beaver population of 5000 in the watershed) and every year or two there are beavers spreading upstream into my pond and doing damage.
You can shoot them early in the AM or at dusk but if there are very many or if you miss they are quick to learn what a menace you are and become terribly wary.
I have had my best luck trapping them with conibear traps. Use #330 traps. You can buy them online along with attractant scent ( from the beavers' castor glands). I usually make small mounds, around 6" high and a foot in diameter at the waters' edge and smear some castor gland extract on the mound. This imitates the way beavers mark their territory. By moving sticks and brush, and with a small amount of shovel work I create a small channel in the shallow water leading to the mound. I place the conibear trap so that it is set in the channel. You can also
find the paths the beavers follow by looking. After they have been in the same spot for a while you can identify their paths in the shallow water and over the dam etc. You can intercept these paths by setting a conibear trap in them. I usually react quickly to beavers because I live on my pond and have valuable trees as part of my landscape. I can't wait for the beavers to start making trails or they will have destroyed my trees,so I bait them. Also, the state trappers take about 4-6 months to respond to requests for animal control and by that time no telling how much damage would have been done.
Beavers are very good to eat, both the backstrap, tendereloin and the haunches. It is like a cross between pork and domesticated rabbit. I have often eaten my game. There is ample online info on how to procede by googling "trapping", "beaver trapping" etc.

Layton Runkle