Detection of light dark matter, such as axion-like particles, puts stringent requirements on the efficiency and dark-count rates of microwave-photon detectors. The possibility of operating a current-biased Josephson junction as a single-microwave photon-detector was investigated through numerical simulations, and through an initial characterization of two Al junctions fabricated by shadow mask evaporation, done in a dilution refrigerator by measuring escape currents at different temperatures, from 40 mK up to the Al transition temperature. The escape dynamics of the junctions were reproduced in the simulation, including the dissipative effects. Inhibition of thermal activation was observed, leaving the macroscopic quantum tunneling as the dominant effect well beyond the crossover temperature.