Nanoscale superconductor/semiconductor hybrid devices are assembled from indium arsenide semiconductor nanowires individually contacted by aluminum-based superconductor electrodes. Below 1 kelvin, the high transparency of the contacts gives rise to proximity-induced superconductivity. The nanowires form superconducting weak links operating as mesoscopic Josephson junctions with electrically tunable coupling. The supercurrent can be switched on/off by a gate voltage acting on the electron density in the nanowire. A variation in gate voltage induces universal fluctuations in the normal-state conductance, which are clearly correlated to critical current fluctuations. The alternating-current Josephson effect gives rise to Shapiro steps in the voltage-current characteristic under microwave irradiation.