The simplest way to soundproof a door is to seal it. By adding foam weather-seal or a weather strip to the door jambs, you are minimizing the chance for sound waves to get through. All air gaps must be sealed on the top and the bottom, which can be done by adding in a floor mounted threshold, a door mounted sweeper or both together, and the cracks around the frame are filled in with a foam sealant. The tighter the seal, the less sound that will escape.
Add another door in front of the first one. A premade frame can be attached to the original frame; make sure the doors open in opposite directions. There should be a two-inch air gap between the doors for maximum sound deadening capability, and the frame must be sealed tightly to prevent any sound from leaking out. Although this is an advanced do it yourself project, it is very effective way to soundproof a door.
Add a sound barrier over the door. There are many types of acoustical sheets, foam, and egg carton shaped soundproofing material that can be glued, nailed or screwed onto a door. The more baffles on the pattern of the material, the greater the soundproofing ability it will possess. These types of acoustical sheets can also be applied to the walls for a symmetrically patterned room for total sound insulation.
Add a door cover panel that is specifically made to cover and soundproof a door. These panels are precut and made to fit specific sized doors. Some types of panels even have hook and loop straps that make mounting simple. For maximum soundproofing, add in another door and cover both of them with an acoustic door covering panel.