We investigate the ionization mechanisms in the Seyfert galaxy Markarian 78, chosen because its narrow-line region (NLR) is dominated by a strong jet-gas interaction. Our principal aim is to ascertain whether or not shock-related ionization is important. We use Hubble Space Telescope (HST) Space Telescope Imaging Spectrograph (STIS) optical and UV spectra from four slits, as well as HST Faint Object Spectrograph optical spectra from 10 apertures. Ultimately, several lines of evidence argue against the importance of shocks but in favor of central source photoionization, even for gas that is clearly accelerated by the jet flow. Following earlier work, we compare emission-line ratios with four classes of ionization model: photoionization of optically thick gas (standard and dusty U sequences), photoionization of optically thick and thin gas (Am/i sequence), and fast shocks. Traditional excitation measures are well matched by all three models, with -3.0 < log U < -2.0, -0.5 < log Am/i < +1.0, and 300 km s-1 < Vshock < 500 km s-1. However, several other line ratios provide clear discrimination, with significantly poorer fits for standard and dusty U models and even poorer for shock models. Importantly, our data span a sufficient range of conditions that they show trends in the ratio-ratio plots, and these trends clearly favor central source photoionization models, particularly the Am/i models. To achieve essentially perfect agreement, we make two slight modifications to the Am/i models of Binette et al.: we lower the ionization of the optically thin component (Mrk 78 has somewhat lower than normal excitation), and we increase the nitrogen abundance toward the nucleus. We suspect that this latter effect has introduced ambiguities in previous work. The UV lines are less helpful than anticipated, partly because of their weakness and partly because high ionization parameter photoionization can mimic shocks. An exception is N V, which, in the absence of abundance effects, can provide unambiguous discrimination. We extend the above analysis to include comparison of line profiles. We first show that even slight differences in velocity between pre- and postshock gas (shock scenario) and optically thin and thick gas (Am/i scenario) predict significant profile differences between [O III] λ5007 and other emission lines. The marked absence of such differences argues strongly against the two-component shock models and also demands, for an Am/i model, that optically thin and thick gas have similar velocity distributions. The possible presence, however, of a weak broad low-ionization component may indicate a minor shock contribution. Further evidence against shock ionization is the absence of correlations between line ratios and kinematic parameters (e.g., FWHM or Vpeak); that is, the calmest and the most turbulent gas both have similar ionization. Conversely, a global decrease in excitation away from the nucleus qualitatively supports central source photoionization. Quantitatively, however, Mrk 78 exhibits the so-called Q problem seen in many other active galactic nuclei, although we speculate that a detailed extension of Am/i-type models may resolve the problem without requiring in situ sources of ionization. Finally, we use NLR excitation and far-infrared luminosity to provide two independent estimates of the nuclear ionizing luminosity. Both these estimates match the simple picture of central source photoionization, yielding the observed emission-line luminosity and a plausible (~20%–100%) NLR covering factor. We conclude that despite the NLR in Mrk 78 being strongly disturbed by radio-emitting ejecta, there is little or no evidence for shock-related ionization. There is, however, strong evidence for central source photoionization, including both optically thick and thin gaseous components. In Paper III we make further use of the STIS data to derive the dynamics of the emission-line region and the nature of the jet flow.