The question is actually quite strange. After all, nails are a dead matter, so they don't need to breathe. Nails grow 'from within': they are fed from the bloodstream. The visible part of your nail - the nail plate - consists of dead cells (keratin (just like your hair) and can't breathe). The cuticle bed and cuticle, on the other hand, obtain oxygen from the bloodstream (and not from the air). All the oxygen that the nail matrix needs to grow is supplied from the blood.