The critical behavior of zero-temperature superconducting transitions which can occur in disordered two-dimensional Josephson-junction arrays are investigated by Monte Carlo calculation of ground-state excitation energies and dynamical simulation of the current-voltage characteristics at nonzero temperatures. Two models of arrays in an applied magnetic field are considered: random dilution of junctions and random couplings with half-ux quantum per plaquette f = 1/2. Abovea critical value of disorder, finite-size scaling of the excitation energies indicates a zero-temperature transition and allows an estimate of the critical disorder and the thermal correlation length exponent characterizing the transition. Current-voltage scaling is consistent with the zero-temperature transition. The linear resistance is nonzero at finite temperatures but nonlinear behavior sets in at a characteristic current density determined by the thermal critical exponent. The zero-temperature transition provides an explanation of the washing out of structure for increasing disorder at f = 1/2 while it remains for f = 0, observed experimentally in supercondoucting wire networks.