In this paper, we explore the integration of two groundbreaking technologies, reconfigurable intelligent surfaces (RISs) and orthogonal time frequency space (OTFS) modulation, with a focus on their use in multicast communications to improve high-speed wireless connectivity. We introduce a phase shift design algorithm for RIS-assisted OTFS systems to maximize the channel gain in the presence of both delay and Doppler spread. Furthermore, we propose a method that utilizes the majorization-minimization (MM) algorithm to jointly optimize beamforming and adjust the phase shifts of RIS elements in a multicast scenario. We validate the effectiveness of the proposed algorithms by numerical simulations. The results demonstrate significant performance improvements over existing benchmark approaches.