We present GHOST, a database of 16,175 spectroscopically classified supernovae and the properties of their host galaxies. We have developed a host galaxy association method using image gradients that achieves fewer misassociations for low-z hosts and higher completeness for high-z hosts than previous methods. We use dimensionality reduction to identify the host galaxy properties that distinguish supernova classes. Our results suggest that the hosts of SLSNe, SNe Ia, and core collapse supernovae can be separated using host brightness information and extendedness measures derived from the host's light profile. Next, we train a random forest model with data from GHOST to predict supernova class using exclusively host galaxy information and the radial offset of the supernova. We can distinguish SNe Ia and core collapse supernovae with ~70% accuracy without any photometric data from the event itself. Vera C. Rubin Observatory will usher in a new era of transient population studies, demanding improved photometric tools for rapid identification and classification of transient events. By identifying the host features with high discriminatory power, we will maintain SN sample purities and continue to identify scientifically relevant events as data volumes increase. The GHOST database and our corresponding software for associating transients with host galaxies are both publicly available.