My guesse either the valve acuating rod on the water inlet end of the long arm with the black ball float needs adjusting or the valve it pushes against needs new washer. Assuming this is the set up.....
On the arm/rod with the black ball float is a small short threaded rod at the opposite end that presses on the valve washer plunger shutting off the water inlet when cistern full. If you release the small nut on the arm/rod and turn the small threaded rod clockwise towards the plunger turn the rod about two full turns then retighten the nut. Allow cistern to fill see if this stops the dripping if not you have to call a plumber / janitor if in a apartment block with one.
BB
April 2011