I’m going to guess that you’ve grown some mold on your basswood. Most likely it hasn’t moved very deep into the pores of the wood. Try letting it dry and sanding the surface. Maybe most of it will sand off. If some remains you can try a bleach solution and see if it that will do it with some light scrubbing with a toothbrush.
The real problem is when mold penetrates deep into the wood. I’ve not found anyway to remove the gray color left behind.
I had this happen to me too. I found that doing as Marty suggested is a good first step. But, I had some pieces that seemed beyond redemption. Rather than give up I tried some hydrogen peroxide, straight up, in a spray bottle to hit the areas that were affected. I let it dry and then sanded again. That made some progress. I went ahead and carved some of the pieces hoping that some of the remaining staining would get cut out, which proved to be mostly true. I went ahead and did my new staining technique, which I posted in another section of this forum. The slightly darker stain made any remaining stains disappear.
That’s not the end of the story though. I had one 10 inch plate that I wanted to use for an anniversary plate for some friends. But, it had some crazy graying in places and I knew the pattern wouldn’t get rid of all of it. So, being a big risk taker (not really), I poured a bit of straight bleach into a small cup and used a medium sized, pointed tip, watercolor brush to “paint” the bleach on the problem areas. Once it dried I gave it another sanding and was pleasantly surprised to see it pretty much gone. I just took a quick look at the mostly carved piece and I can’t tell where the stains were at. I am confident that once it is finished no one will ever know that it had so much mildew on it.
Sounds like you have gotten the best advice. All I could add is Oxalic might help or as you have been told, sodium hydroxide or bleach. If the board is not already thins and off the mildew might be easiest as long as the wood is good and dry.