Bacteriophages ("phages") are viruses that infect bacteria. Since they do not actively self-propel, phages rely on thermal diffusion to find target cells---but can also be advected by fluid flows, such as those generated by motile bacteria themselves in bulk fluids. How does the flow field generated by a swimming bacterium influence how it encounters and is infected by phages? Here, we address this question using coupled molecular dynamics and lattice Boltzmann simulations of flagellated bacteria swimming through a bulk fluid containing uniformly-dispersed phages. We find that while swimming increases the rate at which both the cell body and flagellar propeller are infected by phages, hydrodynamic interactions strongly suppress this increase at the cell body, but conversely enhance this increase at the flagellar bundle. Our results highlight the pivotal influence of hydrodynamics on the interactions between bacteria and phages, as well as other diffusible species in microbial environments.