Lucy
With all due respect, both air conditioners and dehumidifiers use
energy and create waste heat that you will have to get rid of somehow.
A dehumidifier would cause temperature variations in the wine room as
it cycled, the air conditioner would pump heat into the rest of the
basement.

It may be better to find the source of the dampness; it may be that
your rain gutters are not working correctly if you have them. If the
walls are made of block, some homeowners poke holes in the bottom of
the block and build a drip tray around the walls to catch any water
and drain it to a low point or sump pump. The better way is to break
up the floor near the wall and do all of this under the floor if
sealing the wall outside is not a good option.

The mold on the corks looks bad, but is not an issue with a proper
cork. Wines are often stored in damp caves. If mold were an issue,
we would have found that out 200 years ago. I know it sounds silly to
ignore mold growing on the cork, but it is not an issue if the cork is
sound. You just clean it off before opening.

I would fix the water problem anyway, it's not good for the
foundation.
Regards,
Joe