Is it OK to block off an unused root cellar to mitigate moisture problems?
It probably sweats because it isn't vented properly to the outside and blocked from the inside. If it's truly a cold area, it should be vented to the outside and also vapor blocked from the inside. Most likely hot wet air from your house is mixing with cold air in the cold room and condensing on the walls or something.

They deal with these on episodes of Holmes on Homes all the time.  In one episode I think they removed the vents and spray foamed it to be part of the inside house and in another they added vents and a better door to make it separate from the house.