Shower heads typically drip at two points: where the shower head connects to the arm and where the arm connects to the wall. When leaks occur in your shower head, sometimes hand-tightening the shower head or the arm can stop the dripping, but this may not always solve the problem.. Check the threads on both the shower arm and the shower head. If they are worn or cracked, replace the shower head with a new one. If they are not, the leak may be caused by the gasket.