Humidity itself doesn't make clothes smell, slow drying does. Damp fabric sitting for hours lets bacteria build up, and that's the musty smell. The fix is airflow more than heat: hang each piece with space around it, and point a fan at the rack if the air feels still. If the weather won't cooperate, a tumble dryer on low heat works fine for muslin and gets clothes dry before the smell has a chance to start.
How you wash matters as much as how you dry. Pull clothes out of the machine right away. Leaving a wet load sitting in the drum for even a couple of hours in a humid house is enough to set the musty smell into the fibers. Washing muslin on a gentle cold cycle keeps it soft and also keeps loads smaller, since muslin's loose weave doesn't need a packed drum to get clean. Skip fabric softener: it coats the fibers and can trap moisture instead of letting it go.
If a musty smell has already set in, a regular wash won't always clear it. The same damp conditions that cause musty smell can also leave yellow marks on clothes pulled from storage, so store clothes only once they're completely dry, not just dry to the touch. Muslin creases easily and gets softer with every wash, both normal, but neither helps if moisture gets trapped first.