Run a dehumidifier inside the shed. This is an electrical item that removes moisture from the air. They are available in a number of sizes and can be purchased from any good hardware store or online.
Install windows with storm shutters. The windows can be opened to allow for circulation of air, which dries out any condensation. Storm shutters prevent a buildup of condensation on the windows when cold air comes in contact with the warmer shed interior.
Make sure when the shed is built that a proper damp course is installed in the foundation. This prevents dampness from rising up from the ground, which is a common problem in unheated outbuildings.
Apply mastic sealant to the inside rails of the foundation to improve drainage away from the shed.
Lay out packets of moisture-absorbing chemicals. Silica gel and other similar substances can be placed in corners of rooms affected by damp and condensation. These can be bought from DIY stores and are similar to the packets found in shoe boxes.
Detach the walls of the shed and install a raised wooden floor. The raised floor will improve air flow under the shed, which will remove moisture.
Clean the areas affected by condensation with methylated spirits. This will not only stop moisture buildup, but will prevent it in the future if used regularly.