Mortar (or something) falling out of basement brick wall
The white powdery substance is called efflorescence. It's bascially mineral salts leaching out of the masonry. 

The efflorescence itself may not be anything more than a cosmetic problem, however the appearance of efflorescence indicates that there's moisture present in the masonry. This is not entirely surprising--you're talking about an old brick wall below grade.

It may be prudent to monitor the structural integrity of the mortar joints over time to make sure they are not degrading due to the moisture.

The Masonry Institute of America published a nice informative paper on efflorescence, how to avoid it in new construction, and how to remove it from the surface.