High-resolution infrared maps and echelle spectroscopy of the emission from warm molecular hydrogen in the AS 353A/HH 32 outflow are presented. H-2 nu = 1-0 S(1) images are compared with high-quality optical [S II] lambda lambda 6716,6731, H alpha, and I-band images; we find that H-2 line emission ''envelopes'' the HH 32A bow shock, extending downwind beyond its western edge. The H-2 nu = 1-0 S(1) line profiles in HH 32 are all single peaked and Gaussian or triangular in shape. The lines are rather narrow, typically 20-30 km s(-1) (including instrumental smoothing), with peaks that are redshifted by similar to 8-25 km s(-1) relative to the source radial velocity. Although the H-2 data infer low velocities, we nevertheless find that the data are consistent with the high-velocity bow shock models (V similar to 300-350 km s(-1); theta similar to 15 degrees-30 degrees to the line of sight) used to interpret earlier optical data. We also report the discovery of three new Herbig-Haro objects. Designated HH 332, these features are evident only in our [S II] and H alpha images and not in our I-band image (they are not detected in H-2). They are therefore regarded as bona fide HH objects associated with a previously unknown outflow.