What is the purpose of fins on the empennage on some helicopters?
It's a stabilator used on the Blackhawk to reduce attitude changes with airspeed and to offset pitch effects from the tail rotor.  It's part of the automatic flight control system.  

All "traditional" design helicopters have horizontal stabilisers to push the tail down to reduce the nose down pitch in forward flight.  Fixed ones are normally trimmed to provide a more level deck at typical cruise speed.

On the Blackhawk, it's fully movable. It is automatically adjusted according to collective pitch and airspeed. From here, the benefits are:


Align stabilator and main rotor downwash in low speed flight to minimize nose up attitude resulting from downwash.
Provide collective coupling to minimize pitch attitude excursions due to collective inputs from the pilot.
Decrease angle of incidence with increased airspeed to improve static stability.
Provide sideslip to pitch coupling to reduce susceptibility to gusts.
Provide pitch rate feedback to improve dynamic stability.