The image shown below is taken from the structure of NADH bound in the active site of horse liver alcohol dehydrogenase, with DMSO (not shown) as a bound inhibitor. The PDB code for this model is 2oxi. As can be seen from the figure, The carboxamide group on the nicotinamide ring is positioned away from the ribose ring, so we can say that the NADH is bound in the anti- conformation. The carboxamide group is held in place by three hydrogen bonds, to backbone atoms from Val292, Ala317 and Phe 319. In addition to these favorable contacts, alternate conformations of the nicotinamide ring are prohibited by the steric bulk of Thr178, Val203 and the coordinated metal ion (shown as a magenta sphere). If the ring were to rotate by 180 degrees about the glycosidic bond, the carboxamide ring would encounter steric clashes with all of these groups.
The DMSO molecule lies directly above the plane (towards the viewer) of the nicotinamide ring as it is shown in this image. Therefore, we can predict that the pro-R hydrogen is transfered to the substrate from NADH.