Infrared continuum bands that extend over a broad frequency range are a key spectral signature of protonated water clusters. They are observed for many membrane proteins that contain internal water molecules, but their microscopic mechanism has remained unclear. Here we compute infrared spectra for protonated and unprotonated water chains, discs, and droplets from ab initio molecular dynamics simulations. The continuum bands of the protonated clusters exhibit significant anisotropy for chains and discs, with increased absorption along the direction of maximal cluster extension. We show that the continuum band arises from the nuclei motion near the excess charge, with a long-ranged amplification due to the electronic polarizability. Our experimental, polarization-resolved light-dark difference spectrum of the light-driven proton pump bacteriorhodopsin exhibits a pronounced dichroic continuum band. Our results suggest that the protonated water cluster responsible for the continuum band of bacteriorhodopsin is oriented perpendicularly to the membrane normal.