How do you kill beavers?
you just shoot in the head with a fricking 12 gage.
It as illegal to kill beavers in most locations that they are found. To answer the question - you should not kill beavers!
There is a new solution I came across that a company is working on it is called The Beaver Defeater. It will keep beavers from building dams in critical areas such as culverts to stop the damages of Beaver flooding. You can check out the device for yourself at http://www.thebeaverdefeater.com