And welcome to Washington by the way!

Happy to have tried to help. Good luck with the issue.

If it's like that, with cables that looks like those, that is a box for your telephone, internet, cable/satellite, and such. It shouldn't affect your power. Key word is shouldn't. Generally (as in I personally haven't seen any.) Fuse panels will be inside. Some exclusions would be if a large item is outside, most of the time being, hottubs, sanas, ot things like that.

Just trying to determine how much of a problem you may have, and if you do have the singular fuse panel. If the washer and dryer were in the basement then there is a good chance that it would be on the same panel as the basement. But generally they are all in the main breaker box unless whomever had the house before you used equipment with huge power draws (welding, mills, lathes. Just to name a few.). I'm assuming there is no labeling on the door of the panel?

Freezing temperatures shouldn't affect internal wiring, except for drastic circumstances (but I do not claim any amount of expertise, just personal experience.). Is the basement just one large room? Does it have a washer and dryer? Or any other appliances that may run 220/240(usually a double breaker)?

Are there multiple panels? Well that would depend on who made the house and the purpose they had for the rooms. But if you've gone through each breaker and they all check fine, I'd suggest checking any GFI outlets you have sometimes those pop and throw everything off.