I'm with Jason. If you are going to keep that setup, fill it with Great foam and it should attenuate the sound a lot.
The best solution would be to make a wooden stand.
I make them out of 17.5" long sections of 2X4, five per layer ( each layer takes one 8" 2X4. You will need about 10-15 layers, depending on the anvil and your height.).
Lay down five, squirt on five nice lines of glue, and lay on five more at 90 degrees from the first layer. Use a 3.5" deck screw at every cross spot - all 25 of them ( you will need a five pound box of screws). Keep screwing and gluing up the layers until you get to about 2" shy of the height needed ( every layer is 90 degrees from the last). Lay the last layer with the second and fourth boards about 4 feet long. These make great places to lay hammers and tongs , as well as hot blades. They also provide great handles to allow two people to carry the base ( including anvil if not too heavy). If needed, add another layer of the 17.5" boards. The last layer and the long boards can be oak for looks, but the cheap 2X4s work just as well.
Set the anvil on the top and if all is good, draw around the base. When all is right ,apply a good layer of liquid nails where the anvil sits, and set the anvil on. Leave it all as is for a week.
Your best bet for moving anvils and bases around is a good industrial hand truck. Adding wheels to an anvil base is inviting an accident.