As per above, best to check with the authorities for legalities and responsibilities - in most places "nuisance beavers" can be killed.
If the habitat is suitable for beavers then it will be an ongoing problem as young beavers, driven out of their parent's pond will be quick to settle in. Because they are "fur bearing" animals, for a practical solution it might be best to check for a local (licensed) trapper and give him trapping rights.
That being said, 3 acres of the 10 acres I own are part of a 55 acre spring-fed beaver pond and I have to manage the beaver population. A .25 or a .30 PCP will do the job but the problem is that the kill zone on a swimming beaver is only 1 inch high and 2 inches long. Between being in motion (you rarely get a stationary shot) and the angle of the shot, it is hard to get a solid hit and a humane kill - ricochets off the water and off the skull can be a concern.
Trapping is most practical but if you have to shoot them with an airgun (or rimfire) it is best to get some "Beaver Lure" to attract them (beavers are very territorial) so that you can get a good body shot on the beaver when he is out of the water. The lure represents an intruder, put some on a rag (in a weighted plastic) and put it where you can get a clear shot - don't sit too close and mind the wind direction - beavers are at a disadvantage on land, they know that and are wary. Evening is the best time for an ambush. Don't leave the lure when you are not watching as if the beaver discovers it is a fake it will lose its effectiveness. Beavers inspect the dam regularly and the sound of running water attracts them so (carefully) removing a couple of sticks can setup a good shot as well.
To be blunt about it, if you need to get rid of a problem beaver don't play around with rifles - often difficult to get a clear shot and if you spook them they can become very wary (or just become nocturnal) - the most effective solution is a 12 ga loaded with small (#4) buckshot - boom and you are done, saves a lot of time and frustration in a long run. Buckshot is not "over kill" as beaver's head is heavily muscled and the fur/skin is quite thick (for protection - beavers fight a lot) so birdshot is't enough - especially for a big adult (I have weighed several that were over 50 pounds!).
I like watching the beavers in the pond, it's just that my pond is big enough to have 3-4 families (2-6 beavers each) and they can do a lot of damage if I don't manage the numbers.
Oh, and BTW, properly prepared, the young ones are really good eating.
Hope this helps.
Hank