The stochastic generalized Burgers–KdV equations (SGB–KdVEs) arise in the modeling of nonlinear wave phenomena where dispersive, diffusive, and convective effects coexist under stochastic influences. Unlike Brownian-driven models, the incorporation of Lévy noise captures abrupt, non-Gaussian perturbations that more accurately represent realistic wave dynamics. While related studies have mainly focused on Brownian-driven models, the stabilization of SGB–KdVEs with Lévy noise under nonlinear boundary control has not been systematically investigated. To address this gap, we construct a Lyapunov functional and develop a nonlinear mixed boundary controller, from which explicit sufficient conditions are derived to guarantee asymptotic mean-square stability in the presence of stochastic disturbances. To cope with parameter uncertainties, a robust boundary control strategy is proposed, and for systems subject to external perturbations, an H∞ boundary control scheme is further developed to achieve both stability and disturbance attenuation. In contrast to backstepping-based approaches, the proposed method avoids solving kernel equations and offers greater flexibility in handling higher-order nonlinearities and jump disturbances. The theoretical analysis elucidates the coupled effects of noise characteristics, nonlinear terms, and boundary feedback on the evolution of wave energy. Numerical simulations, including scenarios inspired by extreme wave events, validate the theoretical results and demonstrate the effectiveness of the proposed control schemes. Overall, the results establish explicit and verifiable conditions for the boundary control of Lévy-driven stochastic PDEs, ensuring both stability and robust performance.