We investigate the resolved kinematics of the molecular gas, as traced by the Atacama Large Millimeter/submillimeter Array in CO (2-1), of 25 cluster member galaxies across three different clusters at a redshift of z similar to 1.6. This is the first large-scale analysis of the molecular gas kinematics of cluster galaxies at this redshift. By separately estimating the rotation curve of the approaching and receding sides of each galaxy via kinematic modeling, we quantify the difference in total circular velocity to characterize the overall kinematic asymmetry of each galaxy. 3/14 of the galaxies in our sample that we are able to model have similar degrees of asymmetry as that observed in galaxies in the field at similar redshift based on observations of mainly ionized gas. However, this leaves 11/14 galaxies in our sample with significantly higher asymmetry, and some of these galaxies have degrees of asymmetry of up to similar to 50 times higher than field galaxies observed at similar redshift. Some of these extreme cases also have one-sided tail-like morphology seen in the molecular gas, supporting a scenario of tidal and/or ram pressure interaction. Such stark differences in the kinematic asymmetry in clusters versus the field suggest the evolutionary influence of dense environments, established as being a major driver of galaxy evolution at low redshift, is also active in the high-redshift universe.