I think I would leave that for the carvers. In nine years, I don't think I have ever seen a piece from basswood turn up at our turning club. Very soft wood is really not too much fun to turn. You'll run into enough of it when you get some spalted pieces that have some punky spots.
They are balsa but I have stained the basswood before. Its a very soft and porous wood so in my experience it takes stain almost to well. When I dip my rag in stain I make sure to not have any real wet spots. Basically I dip the rag and then fold it over onto its self to make sure the stain is even and not to heavy otherwise it will turn out very dark. I would suggest experimenting on some scrap first if you have that option. Hope this helps
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