I'm not sure how effective it will be, but you could put something under there that might help. Maybe try Dynamat, which is asphalt sheeting used by stereo installers. Rather than rely on adhesives alone, I'd put some staples in it when attaching to the bottom of the seat.
I've seen guys adapt automotive mufflers to four-wheelers to cut down on engine noise. You could fab something up to do the same.
Some of the noise will be coming from the drive system and I'm not sure how you could cut that down.