We use a semiclassical formalism to optimize a microwave single-photon detector based on switching events of a current-biased Josephson junction coupled to a resonator. To detect very rare events, the average time between dark counts τdark should be maximized taking into account that the switching time τsw should be sufficiently small. We demonstrate that these times can be tuned in a wide range by changing the junction parameters, and τdark/τsw ~ 109 can be achieved. Therefore, a junction-resonator arrangement can be used to detect extremely low photon fluxes (e.g., for searching for galactic axions).