If they are common in the area you will not get rid of them by trapping or killing unless you eradicate the species. Much like predators and varmints, if you kill them, others just move in to replace them. Beavers and their dams have been on the earth for decades and served a purpose on waterways in the wild.
They are common on streams and such here in the Ozarks and we are not going to be able to kill them off. Our local biologist said to just keep destroying their dam construction and they will eventually move on to another area.
We wrapped the bottom 3-4' of the larger trees we wanted to protect with standard hardware mesh wire and it stopped them from chewing on them.
They actually did us a favor getting rid of lots of nuisance small saplings along a section of our creek and saved me a lot of chainsaw time. I will say that their one little dam they built actually made me a nice small mouth hole, but it got washed in a flood in the spring. We haven't seen the beavers since.
TAMU Wildlife & Fisheries Sciences