Understanding the mechanisms that lead dense environments to host galaxies with redder colors, more spheroidal morphologies, and lower star formation rates than field populations remains an important problem. As most candidate processes ultimately depend on host halo mass, accurate characterizations of the local environment, ideally tied to halo mass estimates and spanning a range in halo mass and redshift, are needed. In this work, we present and test a rigorous, probabilistic method for assigning galaxies to groups based on precise photometric redshifts and X-ray-selected groups drawn from the COSMOS field. The groups have masses in the range 10(13) less than or similar to M-200c/M-circle dot less than or similar to 10(14) and span redshifts 0 < z < 1. We characterize our selection algorithm via tests on spectroscopic subsamples, including new data obtained at the Very Large Telescope, and by applying our method to detailed mock catalogs. We find that our group member galaxy sample has a purity of 84% and completeness of 92% within 0.5 R-200c. We measure the impact of uncertainties in redshifts and group centering on the quality of the member selection with simulations based on current data as well as future imaging and spectroscopic surveys. As a first application of our new group member catalog which will be made publicly available, we show that member galaxies exhibit a higher quenched fraction compared to the field at fixed stellar mass out to z similar to 1, indicating a significant relationship between star formation and environment at group scales. We also address the suggestion that dusty star-forming galaxies in such groups may impact the high-l power spectrum of the cosmic microwave background and find that such a population cannot explain the low power seen in recent Sunyaev-Zel'dovich measurements.