We use low-dispersion spectra obtained at the Magellan Observatory to study the broad Ha emission from the reverse shock of the infant supernova remnant SNR 1987A. These spectra demonstrate that the spatiokinematic structure of the reverse shock can be distinguished from that of the circumstellar ring and hot spots, even at ground-based spatial resolution. We measure a total dereddened H alpha flux of 1.99(+/- 0.22) x 10-(13) ergs s(-1) cm(-2) at an epoch 18.00 yr after outburst. At 50 kpc, the total reverse shock luminosity in Ha is roughly 15, which implies a L, total flux of neutral hydrogen atoms across the reverse shock of 8.9 x 10(46) s(-1), or roughly 2.3 x 10(-3) M-circle dot yr(-1). This represents an increase by a factor of similar to 4 since 1997. Lyman continuum radiation from gas shocked by the forward blast wave can ionize neutral hydrogen atoms in the supernova debris before they reach the reverse shock. If the inward flux of ionizing photons exceeds the flux of hydrogen atoms approaching the reverse shock, this preionization will shut off the broad Ly alpha and H alpha emission. The observed X- ray emission of SNR 1987A implies that the ratio of ionizing flux to hydrogen atom flux across the reverse shock is presently at least 0.04. The X-ray emission is increasing much faster than the flux of atoms, and if these trends continue, we estimate that the broad Ly alpha and H alpha emission will vanish in similar to 7 yr.