A two-dimensional array of resistively shunted Josephson junctions in the presence of both applied external currents and transverse magnetic fields is studied both analytically and numerically. The dynamics of the system is specified by a set of Langevin equations, and leads to a stationary state, which can be described by an effective Hamiltonian. The latter is then transformed into a Coulomb-gas Hamiltonian, which is, via the renormalization-group technique, shown to exhibit a phase transition similar to that in the absence of external currents: Small external currents merely lower the transition temperature. The effective Hamiltonian further allows us to calculate the current-voltage characteristics, which reveals Ohmic behavior in the low-current regime. We also present results of extensive numerical simulations, which confirm both the nature of the phase transition and the linear current-voltage relation obtained analytically.