We present a microscopic theory of a photon drag effect that appears in a Bose-Einstein condensate of neutral particles, considering indirect excitons in a double quantum well nanostructure under the action of a polarized electromagnetic field. It is shown that the dynamical polarization of excitons results in a resonant behavior of the exciton photon drag flux when the frequency of light is close to the gap between two energy levels of internal exciton motion. Specifically, we consider the ground and first excited energy states characterized by the angular momentum difference ±1, and thus, the helicity of light matters. We show that the resulting drag current is caused by both Bose-condensed particles and the particles in the excited states. As a result, the total current represents a superposition of thresholdlike and resonant contributions - a property which can be used in frequency-selective photodetection.