Fortunately, nose blindness only affects one of your senses, and you don’t need your nose to verify a basement with a musty smell.
Mustiness is caused by mildew and mold, which — for better or for worse — your eyeballs can easily detect. Do a careful inspection of your basement, from the darkest corner to the surface of every cardboard box or bookshelf. If you find gray or white splotches anywhere, it’s probably mildew. If it’s fuzzy, (oh no!) it’s mold.
First, you’ll want to bust up those existing odors. Then, you’ll want to make sure they never return. Some elbow grease with regular household cleaner will scrub away mildew. Bleach isn’t the cure-all for mold. If often can exacerbate the problem.
To prevent mildew and mold from returning, consider running a dehumidifier or improving air circulation and sunlight exposure in the affected area if possible.
For chronic mustiness, you can deodorize rooms by setting out bowls of vinegar, cat litter, baking soda, or — as crazy as this sounds — an onion also will do the trick. Cut one in half and let it sit in a bowl in the room. The onion smell goes away in a few hours, and so will the dankness.
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