The effects of dissipation are studied in the proximity of a quantum phase transition. The prototypical system is a resistively shunted two-dimensional Josephson junction array, simulated by means of an advanced Fourier path-integral Monte Carlo algorithm. A reentrant superconducting-to-normal phase transition is the most dramatic signature of the strong nonlinear fluctuations characterizing the quantum critical region at finite temperature. It is shown that a region of quantum criticality exists at finite dissipation and the zero-T phase diagram is constructed. The latter is of crucial importance for the fabrication of samples with interaction parameters suited to match the quantum critical region. It is shown how quantum fluctuations are quenched by dissipation, leading to the removal of the reentrance. As dissipation can drive the system in and out of the critical region, it reveals as a fundamental aspect for both theoretical and experimental investigations.