There is a strong chance that your neighbor's possessions contain mold growth if he or she has laid cardboard boxes or other biodegradable materials directly on the floor or up against the wall. To prevent mold, personal goods should be stored on plastic or metal shelving, away from the wall and up off the floor. There may also be a leak in that area of the basement that is contributing to the spread of household mold.
This isn't your storage area, but the air you breathe while in the basement is being affected by the mold; in addition, up to a third of the air in a small residential building can come from the basement, due to the stack effect (warm air rises).
In a situation like this where mold is suspected in basement storage, it's best to speak to your neighbor as well as to the building management about this problem, to see if you can get the space cleaned up by a mold remediation professional. Any leaks that are present should be repaired, and the basement should be dehumidified (with the relative humidity less than 50%) in the warmer months to prevent the return of mold.