MDF is a good acoustic material... but I'd be inclined to use pressure-treated plywood (or at least exterior-grade regular plywood) in a door like that. Any moisture that gets in there, won't affect the plywood... however, MDF will swell up and disintegrate, over time, if it gets wet.
Most doors won't get wet, to much of an extent... but I like to be safe rather than sorry...
Regards,
Gordon.