We theoretically study how the phase of a light plane wave propagating in a resonant medium under electromagnetically induced transparency (EIT) is affected by the uniform motion of the medium. For cuprous oxide (Cu2O), where EIT can be implemented through a typical pump-probe configuration, the resonant probe beam experiences a phase shift (Fresnel-Fizeau effect) that may vary over a wide range of values, positive or negative, and even vanishing, due to the combined effects of the strong frequency dispersion and anisotropy both induced by the pump. The use of such a coherently driven dragging medium may improve by at least 1 order of magnitude the sensitivity at low velocity in optical drag experiments.