It is vital to follow proper safety procedures when dealing with insulation -- let alone if it is moldy. Wear safety glasses, a respirator mask, long sleeves and gloves. Fiberglass insulation contains fiberglass which can cause cuts and irritate lungs if inhaled in small particles. You should also avoid breathing harmful mold spores. The area will need to be protected as well. Lay down plastic drop cloths underneath where you will be working. Having a few large garbage cans nearby helps make it easier to dispose of the wall material and the insulation.
The wall in front of the insulation must be removed first. Cut through the drywall with a utility knife and begin prying it away from the wall. While you can break through it with a hammer to tear it off in chunks, using a utility knife makes the job easier. Remove any screws or nails in the wall joists and clean up any drywall bits that may still be attached to the wall framing.
Roll up the insulation. If it is badly rotted, the insulation will just come out in chunks. However, if you can, roll it up and discard it immediately. If the insulation was blown in, this will take more time to clean out. Use a dust pan to help scoop it up and toss each pan full into garbage cans as you work. Make sure all the insulation is removed so no small pieces will be left to harbor mold.
The area underneath the insulation should be sterilized and cleaned of any residual mold or mildew. Use bleach and water and a long-handled scrub brush to clean the area. Let this dry completely. Place a fan nearby to help before you install any new insulation or wall material. This should completely remove the problem. To prevent future occurrence of mold, use a vapor barrier between the concrete wall of the basement and the insulation.