Mike,
Be happy you have this problem... After a co-worker stopped by shop and saw my set up, he decided to make like me and went out and bought the first anvil he stumbled onto. Now I explained how to check an anvil for cracks, for hard face, response, sway-back, etc... but the bug bit him good. He was so proud of his new "anchor" he just had to bring it by to show me. I could see right a away he bought junk, the anvil had the surface of an old "black top road" and the metal looked the color of old lead and the anvil had about as much response as soft packed earth and it "sang" with about that much "Thud..."
So the moral is, you've got a good anvil and be thankful you have this problem...
I have a 130 pound Sliggo with a hard face that rings bad enough to crack class. Since I also live in a very residential setting with a very strict home owners association, I have to be very careful not to piss anyone off. I have a three car garage with the third stall completely enclosed, walled off and sound proffed, but the forging goes on in the backyard. I mounted the anvil base, a tree trunk, in the ground 2 feet, I leveled the anvil feet and the base, I installed some very hard rubber on the base to cushion the anvil, not to absorb the hammer blows, but to stop the vibration (like when you touch a tunning fork, it stops vibrating), and I hang a steel ball (15 lb) by a section of inner tube from the horn or beak if I'm just using the face. This may seem pretty elaborate, but I like my neighbors and it would sure piss me off to be stuck listening to some fool pounding on a leaf spring.... Remember, they may not care what comes off your anvil, just what come off their TV set.
I also use a Makel two burner, blower forge and if you've every used one of these, then you know they sound like a small jet engine. I often sacrifice a hot dog or two to make my neighbors think I'm BBQing...now I didn't say they were smart, I just said they don't care for the noise.
And I absolutely agree with TJ concerning the ventilation. Remember, you burn off about 1% of the metal every time you bring the metal to color and that's a lot of heavy metal to be breathing...
Well, which ever way you go, remember, it's the vibration that makes the noise so anything you do to stop that will quiet your hammering...
Good Luck