I recently built a 10 x 10 storage shed.  I used standard 2x4
construction, and OSB walls and floor.  It's been fairly rainy lately,
and although the roof is solid (there are no water leaks) the
resulting damp air is causing surface mold on the OSB inside the shed.

My question is how can I prevent this?  I realize that I can paint
everything which will solve the surface areas, but what about the
sections of the OSB that are flush with the 2x4s (and subsequently I
cannot paint the surfaces in between).  These sections that are touching
other wood appear to have the most mold, presumably due to the reduced
airflow.

I have two air vents, and window and a ridge vent on the shed.

Supposedly vinegar, bleach, ozone, and uv light kill mold.

More vents - and generate an air flow - in at the bottom, out at the top.