We study the influence of a tunable dissipative environment on the dyn
amics of Josephson junction arrays near the superconductor-insulator t
ransition. The experimental realization of the environment is a two di
mensional electron gas coupled capacitively to the array. This setup a
llows for the well controlled tuning of the dissipation by changing th
e resistance of the two dimensional electron gas. The capacitive coupl
ing cuts off the dissipation at low frequencies. We determine the phas
e diagram and calculate the temperature and dissipation dependence of
the array conductivity. We find good agreement with recent experimenta
l results.