One core goal of the Kepler mission was to determine the frequency of Earth-like planets that orbit Sun-like stars. Accurately estimating this planet occurrence rate requires both a well-vetted list of planets and a clear understanding of the stars searched for planets. Previous ground-based follow-up observations have, through a variety of methods, sought to improve our knowledge of stars that are known to host planets. Kepler targets without detected planets, however, have not been subjected to the same intensity of follow-up observations. In this paper, we constrain better the stellar multiplicity for stars around which Kepler could have theoretically detected a transiting Earth-sized planet in the habitable zone. We subsequently aim to improve estimates of the exoplanet search completeness-the fraction of exoplanets that were detected by Kepler-with our analysis. By obtaining adaptive optics observations of 71 Kepler target stars from the Shane 3 m telescope at Lick Observatory, we detected 14 candidate stellar companions within 4 '' of 13 target stars. Of these 14 candidate stellar companions, we determine through multiple independent methods that 3 are likely to be bound to their corresponding target star. We then assess the impact of our observations on exoplanet occurrence rate calculations, finding an increase in occurrence of 6% (0.9 sigma) for various estimates of the frequency of Earth-like planets and an increase of 26% (4.5 sigma) for super-Earths and sub-Neptunes. These occurrence increases are not entirely commensurate with theoretical predictions, though this discrepancy may be due to differences in the treatment of stellar binarity.