I am unclear on what a Basswood Falls Quetico permit entails. Do you have to cross the falls portages all the way to Crooked, or do you just have to go in that direction and camp near the falls in Basswood?

In the past it's been labeled a "Basswood River" permit, not Basswood Falls. Technically to fulfill your permit obligations you have to travel on the Basswood River, taking a direct route to the river. You are obligated to paddle on the river but you are not required to paddle the entirety of the river to Crooked (you could leave the border via Greer (an interesting way to get to Kett) or you could portage into the Basswood River and turn around and head into North Bay- though the ranger might give you a bit of grief about this plan).