Before you put anything away, wash it cold on a gentle cycle and get it fully dry. Hang it in the sun or use a low tumble setting (you can tumble dry muslin baby clothes without any trouble). Skip the fabric softener and skip high heat. Any dampness left in the fibers, even a trace you can't feel, turns into a musty smell once a piece sits sealed away for months. Muslin's open weave dries faster than denser fabric, which helps, but it still needs to be bone dry, not just damp-dry, before it goes in a box.
Skip sealed plastic bins and vacuum bags if you can. In a humid place, sealed plastic traps whatever moisture is already in the bag with the clothes, and that's exactly what breeds mildew. A cotton storage bag or a fabric-lined box with a loose-fitting lid lets some air move through instead. If you do use a plastic tub, drop in a few silica gel packets and keep it up off a concrete floor, where damp tends to collect. We cover why humid air causes that musty smell in more detail if you want the longer version.
Open the box every few months rather than leaving it sealed until the next baby arrives. A quick check lets you catch damp or musty pieces early and rewash and re-dry them before the smell sets in properly. When the next baby does come along, give everything a wash again even if it looks fine, since fabric that has sat for a year or two can pick up a flat, stored smell even when it was stored well. If you're weighing whether to keep what you have or buy fresh for round two, we've written about new muslin clothes against hand-me-downs for a second baby.