I have wooden oak floors no problem for 23 years. Just finished my basement and now on this cold 10° night I noticed the oak floor is sweating. Any suggestions what is causing it and how to fix it?

Is the basement heated? Is the floor cold enough to cause condensation?

I'd guess that whatever moisture your basement has always had (from the ground under the floor, or appliances) got out through windows, sills, etc, and now you have sealed off those avenues of escape, so the wood floor is what is left as the air rises. But it does seem odd that it would condense on top of your floors above, where it should just keep rising and getting out through windows and doors. Vapor condenses when it hits cold.
The typical solution for basement dampness is a dehumidifier with a drain so you don't have to empty it by hand.

Tell me more. Does the moisture go away or does it get worse? Back to you, Tom