The Cl 1604 supercluster at z ~ 0.9 is one of the most extensively studied high-redshift large-scale structures, with more than 500 spectroscopically confirmed members. It consists of eight clusters and groups, with members numbering from a dozen to nearly a hundred, providing a broad range of environments for investigating the large-scale environmental effects on galaxy evolution. Here we examine the properties of 48 post-starburst galaxies in Cl 1604, comparing them to other galaxy populations in the same supercluster. Incorporating photometry from ground-based optical and near-infrared imaging, along with Spitzer mid-infrared observations, we derive stellar masses for all Cl 1604 members. The colors and stellar masses of the K+A galaxies support the idea that they are progenitors of red sequence galaxies. Their morphologies, residual star formation rates, and spatial distributions suggest that galaxy mergers may be the principal mechanism producing post-starburst galaxies. Interaction between galaxies and the dense intracluster medium (ICM) is also effective, but only in the cores of dynamically evolved clusters. The prevalence of post-starburst galaxies in clusters correlates with the dynamical state of the host cluster, as both galaxy mergers and the dense ICM produce post-starburst galaxies. We also investigate the incompleteness and contamination of K+A samples selected by means of Hδ and [O II] equivalent widths. K+A samples may be up to ~50% incomplete due to the presence of LINERs/Seyferts, and up to ~30% of K+A galaxies could have substantial star formation activity.