Contact the Department of Natural Resources in your state to ensure you're allowed to remove a beaver dam on your own. Some states require you to have a permit while others will require a government official to inspect the dam first.
Remove the beavers from the area. Most states request that you humanely euthanize the creatures by shooting them. If this is not possible, you can contact animal control and request that they kill or capture the beavers and relocate them to another area.
Contact any neighbors in the area who will be directly affected by the removal of the dam. This is often required before you're allowed to remove the structure. In many states, if you remove a beaver dam on your property and it floods your neighbor's yard, you will be required to pay to fix the damage. Make sure all parties understand this before moving forward.
Contact a licensed blaster to use explosives to remove the dam. All blasters must have a permit and the signed consent of the landowner before performing the blast. If you would rather not use explosives, you may also remove the dam by hand using a shovel, pick or backhoe. This method is often safer, but will take more time and not fully remove the structure.