On 2019 August 14, the LIGO and Virgo Collaborations detected gravitational waves from a black hole and a 2.6 solar mass compact object, possibly the first neutron star-black hole merger. In search of an optical counterpart, the Dark Energy Survey (DES) obtained deep imaging of the entire 90% confidence level localization area with Blanco/DECam 0, 1, 2, 3, 6, and 16 nights after the merger. Objects with varying brightness were detected by the DES Pipeline, and we systematically reduced the candidate counterparts through catalog matching, light-curve properties, host-galaxy photometric redshifts, Southern Astrophysical Research spectroscopic follow-up observations, and machine-learning-based photometric classification. All candidates were rejected as counterparts to the merger. To quantify the sensitivity of our search, we applied our selection criteria to full light-curve simulations of supernovae and kilonovae as they would appear in the DECam observations. Because the source class of the merger was uncertain, we utilized an agnostic, three-component kilonova model based on tidally disrupted neutron star (NS) ejecta properties to quantify our detection efficiency of a counterpart if the merger included an NS. We find that, if a kilonova occurred during this merger, configurations where the ejected matter is greater than 0.07 solar masses, has lanthanide abundance less than 10-8.56, and has a velocity between 0.18c and 0.21c are disfavored at the 2σ level. Furthermore, we estimate that our background reduction methods are capable of associating gravitational wave signals with a detected electromagnetic counterpart at the 4σ level in 95% of future follow-up observations.