I am going to freely admit that I typed the name into behindthename thinking "This is a waste of time, this name doesn't exist." When not one but two options popped up (one Polish, one Hebrew), I was shocked!
Nope, I've never heard this name before. It does seem to be a nickname for other names, but not being Polish or Jewish, I can't really make a judgment call on how popular Basia is over the longer forms.
I'm still not sure I like it yet, but I won't hate it on it at all!