The verb technically appears to be to damp. FumbleFingers notes the OED's definition. M-W and the Random House Dictionary (at Dictionary.com) under dampen merely refer the reader to the relevant definition of damp. The things in a piano are called dampers, not dampeners. In a more formal context, damp is the proper choice.
Dampen does seem to enjoy use. On the Wikipedia page for damping (music), there are instances of dampening. NOAD lists as a subsidiary definition, "reduce the amplitude of (a sound source)." M-W and Random House both acknowledge that it has the same meaning in music as to damp. In an informal context, dampen is fine.