We present a detailed morphological analysis of the galaxy populations in the first two clusters to be completed in an extensive observational study of nine high-redshift clusters of galaxies. These two clusters, Cl 0023+0423 and Cl 1604+4304, are at redshifts of z=0.84 and z=0.90, respectively. The morphological studies are based on high angular resolution imagery taken with Wide Field Planetary Camera 2 aboard the Hubble Space Telescope. These data are combined with deep, ground-based BVRI photometry and spectra taken with the Keck 10m telescopes. The morphological classifications presented in this paper consist of two parts. First, we provide a quantitative description of the structural properties of ~600 galaxies per cluster field using the Medium Deep Survey automated data reduction and object classification software. This analysis includes the galaxy position, photometry, and best-fit bulge+disk model. Second, for the brightest subsample of ~200 galaxies per cluster field, we provide a more detailed morphological description through a visual classification based on the revised Hubble classification scheme.