i'd guess it's a combination of probs. you might not be able to cure the sympathetic vibrations, but you should be able to stop it being a problem:
change to a different weight of strings
adjust the truss rod (You might have to play with this to get a sweet spot where it is playable, and there is no sympathetic buzzes or direct fret buzz)
marginally raise the nut and or bridge
drop the fret height (not really recommended)
to try and reduce the sympathetic vibrations dampening could help a little, if overall harmonic vibration is causing the sympathetic vibration or contributing to it. it depends on how the main energy is being transferred to the string:
through the air
through a mechanical linkage (more likely)