Drywall is a porous material. Its core is gypsum, and its outside is paper, so a mold infestation can be deep and impossible to remove. You can't cover mildew with paint because the mold will continue to grow underneath. In the best case scenario, the paint will bubble over time. In the worst case scenario, the mold will travel across the drywall and to other areas of the room, leading to a serious infestation.
The best approach is to use a utility knife to cut out the moldy drywall. If the affected area is more than several square feet, it might be necessary to hire a professional. Before beginning the project, open any windows in the room that can bring in fresh air, but cover any doorways or openings to other areas in your home as you risk allowing the mold spores to spread elsewhere. Wear a mask that can block mold spores. Dampen the moldy area with water to minimize the spread of airborne spores. Score the area around the mold with the utility knife, then pry out the bad section. Seal it tightly in a garbage bag and take it outdoors.
If the drywall has a glossy paint finish that the mold has not yet penetrated, it might be possible to clean off the mold. For relatively clean surfaces, a solution consisting of 1/4 to 1/2 cup bleach per gallon of water should be sufficient to disinfect the area. Use as much as 1 1/2 cups of bleach per gallon for dirty surfaces. Wear rubber gloves, and allow the solution to sit on the surface for at least 15 minutes for maximum disinfection. You can't always see mold spores, so it's best to clean the entire room with a disinfectant, including ceilings and floors. Proper ventilation is important, so keep the windows and doors open.
Prevent future mold growth by decreasing the humidity of the area. Dehumidifiers are effective, but ensuring adequate and ventilation by frequently opening windows and doors also can work. Special forms of drywall exist to inhibit mold growth in areas with frequent, extreme humidity, such as bathrooms, laundry rooms and basements. Mold-resistant drywall is available at most retail hardware stores, but it often is more expensive than standard drywall.