This article presents the optimization of hydraulic turbines for the suppression of cavitation, defined as the formation of vapor bubbles when pressure drops below the vaporization threshold and their subsequent collapse in higher pressure regions. In modern hydraulic turbines, cavitation remains a major concern, as their extended operating range driven by strongly varying energy demand, results in operating conditions with large pressure variations. Cavitation must be addressed during the design, as it leads to efficiency deterioration and material erosion. An efficient tool for cavitation mitigation is the adjoint method, which provides sensitivity information with respect to shape variations. Previous studies using continuous adjoint formulations for single-phase flows have successfully optimized Francis and Kaplan turbines by increasing blade surface pressure, thereby indirectly reducing cavitation. The present work employs a continuous adjoint formulation for two-phase flows and extends it using three different cavitation models. Embedded in an optimization loop, the method explicitly accounts for cavitation and captures the flow behavior even when cavitation is not fully suppressed, a condition commonly encountered at the limits of the operating range. The formulation includes the adjoint to the Spalart-Allmaras turbulence model equation, essential for accurate sensitivity prediction, and employs a computationally efficient strategy, that solves an additional adjoint equation to avoid computing geometric sensitivities in the entire domain. The method is developed within an in-house GPU-enabled CFD solver based on a vertex-centered finite-volume discretization and is applied to the shape optimization of a Kaplan hydraulic turbine, targeting cavitation minimization at two distinct operating points while retaining efficiency. The resulting optimized shapes using the three different cavitation models are compared. The design space comprises the blade geometry, described by two B-spline surfaces, as well as the hub and shroud generatrices, represented by Bézier curves, while geometric constraints on blade tip clearance, leading edge tangency and trailing edge thickness are enforced.