A fresh coat of paint can give any room an instant facelift, transforming the space into something totally new. Before you being painting, make sure the walls are in their best possible state. Damp walls could be a sign that your house has a range of problems. You still can paint such walls; you just need to address the source of the dampness before you begin. Otherwise, the paint won't dry properly and will streak.
Call a professional handyman or woman. The problem could be caused by clogged gutters, debris in external pipework, a damp cellar, or gas appliances that are not vented or improperly vented. Since your safety may be at stake, let a professional assess the situation and fix the dampness. Don't use gas appliances until a professional has fixed them or given them clearance.
Shut the windows of the room you plan on painting. Plug in a dehumidifier and place it in the centre of the room. Let it run according to the manufacturer's instructions.
Tape off the baseboards and mouldings as well as any other areas that you don't want paint on, such as windowsills.
Apply primer to the walls as you normally would. Allow it to dry. Apply two coats of paint, allowing the paint to dry between coats. Once the final coat of paint is dry, apply a fungicidal sealant. The sealant is a precautionary measure since you started off with damp walls.