Unless you have an unusually large basin wrench there is usually an angle where it can fit on and turn at least a little bit. Generally that is enough to loosen the nut. Yes, it is is PITA removing a nut 1/8th of a turn at a time, but it can be done.
Another option might be to make a stack of tubular wrenches (the kind used to remove shower stems from the wall). Whatever is sticking through that nut would run down the middle and the stack could extend below the bottom of the sink so that you could put a plumber's wrench on the outside.
There is something called a "claw foot" which you might be able to use. It is employed with a long socket set extension and a ratchet at the other end.
Lastly, in the picture it looks like that nut may have 4 flanges sticking out from under it. If those really are part of the nut, or are part of a flat washer under it, you might be able to place a flat bladed screwdriver so that you could tap the side of one of those flanges and break the nut loose that way.
