The initial interaction of a supernova with its surrounding medium gives rise to a double shell structure bounded by shock waves, in which the deceleration of the supernova gas is subject to hydrodynamic instabilities. For the case where the surrounding medium is a circumstellar wind, the high density at early times can give rise to radiative cooling of the shocked supernova gas, in which case the shocked supernova gas forms a very thin shell. We analyze the instability in the thin shell approximation and find an expression for the linear growth rate that can be compared to results from adiabatic calculations. For perturbed wavelengths that are larger than the adiabatic supernova shell thickness, there is excellent agreement between the two growth rates; at smaller wavelengths, the thin shell growth rate becomes larger than that in the adiabatic case. The results show that the thin shell analysis incorporates the salient physical situation for the instability. Numerical simulations confirm the instability analysis in the linear regime and allow the instability to be studied in the nonlinear regime. With radiative cooling, the instability saturates at an amplitude comparable to that found for the adiabatic case - at about half the shell thickness for one case. It appears that once the density contrast becomes large, the outcome of the instability no longer depends on the density contrast. The importance of the instability for radio supernovae is that cool, partially ionized gas is mixed into the region that may give rise to radio synchrotron emission. The resulting free-free absorption may be a factor in the turn-on of radio supernovae. Variations in the shell column density around the shocked shell can be a factor in the absorption of X-rays from the reverse shock front. The clumpiness can also give fluctuations in the emission-line profiles of shocked ejecta gas.