Only two things I can think of.
Is it possible that the basement litter box may be really, really cold or drafty? Especially if it’s sitting on bare concrete? Is it sitting up on a piece of carpeting or several layers of cardboard, or is it sitting directly on the floor?
My other thought was thinking that perhaps the basement litter box was a smaller size or a different color than the other one? Or perhaps not kept as secluded – or perhaps too secluded?
Also, I’m assuming the litter boxes are just plain litter boxes, no rims or tall lids on them? You might have greater success with the biggest litter box you can find, that has a tall, walk-in lid on it. Or, you might have great success with using a cement-mixing tub from Home Depot or Lowe’s, instead; the sides are tall enough that pooping over the edge will be close to impossible. Plus, the tub is big enough that if the cat is fussy about both pooping and peeing in the same spot, they can choose opposite corners (or separate boxes) for such activities.
Useful advice at: http://www.msnbc.msn.com/id/19859148/
Good luck!