Dry the antique furniture. If the furniture resides in a musty garage or basement, move it to a drier environment and remove the moisture from the furniture by running a portable dehumidifier or by turning on a heater to reduce the humidity in the air. Turn on fans facing the furniture to increase air circulation and speed the drying.
Vacuum the furniture to draw away the dry mildew. Attach a thin hose extension to the vacuum and use a gentle suction. For best results, use a HEPA vacuum, which is specially designed to absorb impurities. If you do not have a HEPA vacuum, clean the vacuum filter outdoors immediately after use.
Dampen a sponge with hydrogen peroxide and blot areas where mildew stains appear. Peroxide is a natural and safe alternative to bleach, effective in neutralizing mildew. While it will not harm fibers of antique furniture, it may ruin certain finishes, so perform a spot test before applying peroxide to large areas of the object. Blot gently in an inconspicuous area, and if no discoloration occurs, continue spot treating the antique furniture.
Store the antique furniture in a dry location. If you have high indoor humidity but you want to keep the furniture in your home, keep humidity under control. Run a dehumidifier or a heater during the day. Since wood can provide food for mold, keep the furniture dry at all times.