(As a harpsichord and early music enthusiast, I say let the tracker action speak. It adds both percussive effects and insight into the production of sound, especially lower notes, where the sound perceptibly lags behind the keypress.... But, that said:)
Choice and placement of microphones may be your primary means beyond your idea of damping sound at the keyboard. Some dynamic mics, such as the Electro Voice RE-20, have excellent audio frequency range for close sounds, can handle very loud sounds (they are often used for trumpet and sax in jazz recordings), yet attenuate sounds greatly as the distance between the sound source and the microphone increases. The latter is one reason it is popular in radio broadcast production. Because it is dynamic, it doesn't have the proximity effect of condenser mics.
The down side is its cost and its need for a superb recording chain, owing to its need for pre-amplification (it's not, in other words, a "hot" mic). There is a younger brother, the RE27N/D, that is 6 dB hotter, but even more expensive.
Along with selection of a mic with relatively low sensitivity, very close placement of the microphone(s) to the pipes themselves will much reduce any sound not at the pipe rank itself.
Dave