Understanding the Noise
Most countertops are made of laminate, tile or a solid surface material like Corian brand. The construction of a laminated countertop is particle board with laminate affixed to the surface and secured to the top of the cabinets. When using a KitchenAid mixer on such a surface, the sound is actually amplified due to the open area of the cabinets. A buffering or more solid material must be placed under the mixer to block some of the vibration.
Granite, marble and quartz countertops are more solid materials and will not create as much noise as the laminates. Keep in mind that the KitchenAid mixer has about a 325-watt motor and weighs around 27 pounds; it is a commercial-based product that was introduced to the home kitchen.
Blocking all the noise from a mixer this size is impossible, but you can reduce some of the noise with a few modifications. A simple step is to run your mixer on a slower speed if your recipe allows.
Creating a Buffer
If you don't have a granite, marble or quartz countertop you can buy a piece in the appropriate size to place under your mixer. Go to a granite company for a scrap piece and see if they will cut it to size and smooth the edges for you. You can also place self-adhesive rubber stoppers on the bottom of the piece of granite to add more of a buffer and keep it from slipping. If your KitchenAid mixer doesn't come with rubber stoppers on the bottom, you may want to add some to it as well.
Another inexpensive option is to buy a butcher block that is at least one inch thick. You can find these at DIY stores or kitchen suppliers, or you could even make your own by gluing sanded pieces of maple wood together. Again, add rubber pads if it doesn't already have them on the bottom. This will prevent slipping and scratches on your countertop and create an air pocket between the counter and the butcher block. The vibration--and hence the noise--will be considerably reduced.