Brick Shed - Mildew problems

Hi

I have a brick shed which I cleaned out, repaired and fitted worktops inside of a few months ago. So now instead of a dank musty damp space with water penetration and no storage possibilities I now have a (comparatively) nice space for working in.

However I do have some troubles remaining as some of the wood and fabrics have mildew on them, but most of them are ok. Is there anything I can do to prevent this sort of damage? Also is it reasonable for me to expect to store cardboard boxes in the shed without them becoming flimsy and/or unusable or am I expecting to much?

Thanks