Several years ago I had my circa 1870 rock cellar walls sprayed with gunite concrete on the inside. The contracter also put down a layer (4-6") of gunite on a small remaining section of dirt floor. The difference in the cellar moisture level after covering such a small space was hard to believe.

Anything you can do to prevent moisture migration from the dirt to the cellar will help. Six mil plastic would work but would have to be lapped up onto the foundation walls and sealed with roofing cement or something similar. A similar lapping and sealing would be needed against the wooden walls, but here a product like Ice and Water Shield lapped down over the plastic would probably be better. I think I'd also want to check the wood walls carefully for rot in this kind of environment before doing anything with them.