Unfortunately, the shed may be too cold, while the basement will be too warm for dormancy. I would place it in the shed, but insulate the tree and pot so it doesn't freeze. I don't have experience with this, but there should be some good info if you search it. Good luck!

It seems kind of pointless to dig it out of the ground only to put it in an outdoor unheated shed. It shouldn't have any dormancy problems in the basement since it is significantly less light.

To successfully overwinter a semi-tropical tree, like the fig tree, you must provide a temperature range of about 45 degrees, or less...but, above freezing... around the roots, and just enough hydration so that the tree doesn't totally dry out. Roots kept in this temp range will keep the trees dormant, and while dormant, light will have no effect on the tree.

Your shed is too cold and your basement is too warm for overwintering. A very cool, attached garage might be an option. Your goal is to keep your tree dormant, but alive until Spring weather will support the new growth. You basement will not allow your tree to go dormant and the new growth will be weak and unhealthy because of the lower light levels. This growth will probably be lost once the tree goes out for the Spring-'13 growing season.

On this forum, this very subject has been covered, and a search will give you many options, ranging from the very simple, to more elaborate set-ups for controlling temperatures...like heat mats, and thermostatically controlled heat-cables.

I wish you luck. You could always bury your trees about 18" deep, and pile a good layer of snow and mulch over the buried trees for insulation. Then dig the trees up next Spring. Just a thought.

Frank