This paper presents chemical abundance measurements for 37 damped Lyα systems at z > 2.5 observed with the Echellette Spectrograph and Imager on the Keck II telescope. We measure the H I column densities of these damped systems through Voigt profile fits to their Lyα profiles, and we implement the apparent optical depth method to determine ionic column densities. Figures and tables of all relevant data are presented. A full analysis of the chemical enrichment history described by these observations will be presented in a future paper. This data set is also valuable for efficiently planning future echelle observations and for rough abundance pattern analyses. We aim to make this entire data set public within 3 years of this publication.