The typical beaver traps usually kill them very quickly. THe problem with eradicating beavers is that they learn very quickly. I would get some one in there hwo traps a lot of beaver each year and have him get as many as he can. Any that become trap shy would then be shot. I advise getting in a tree stand in full camo, and not moving at all. It has always amazed me how quickly thiose things learn what is dangerous-man with gun, trap, whatever.
I have seen live traps for beaver on TV. I am sure you could find them on the net, but I see no need. Beavers are swimming rats, as I usually call them.
I the past I did a lot of beaver damage control trapping and shooting. Beavers are just like everything else in the animal world-deer, wolves, geese, whatever. THey look so cute on TV, but when you are overrun by them in your own yard, their cuteness quickly evaporates!