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Stay cables exhibit both great slenderness and low damping, which make them sensitive to resonant phenomena induced by the dynamic character of external actions. Furthermore, for these same reasons, their modal properties may vary significantly while in service due to the modification of the operational and environmental conditions. In order to cope with these two limitations, passive damping devices are usually installed at these structural systems. Robust design methods are thus mandatory in order to ensure the adequate behavior of the stay cables without compromising the budget of the passive control systems. To this end, a motion-based design method under uncertainty conditions is proposed and further implemented in this paper. In particular, the proposal focuses on the robust design of different passive damping devices when they are employed to control the response of stay cables under wind-induced vibrations. The proposed method transforms the design problem into a constrained multi-objective optimization problem, where the objective function is defined in terms of the characteristic parameters of the passive damping device, together with an inequality constraint aimed at guaranteeing the serviceability limit state of the structure. The performance of the proposed method was validated via its application to a benchmark structure with vibratory problems: The longest stay cable of the Alamillo bridge (Seville, Spain) was adopted for this purpose. Three different passive damping devices are considered herein, namely: (i) viscous, (ii) elastomeric, and (iii) frictions dampers. The results obtained by the proposed approach are analyzed and further compared with those provided by a conventional method adopted in the Standards. This comparison illustrates how the newly proposed method allows reduction of the cost of the three types of passive damping devices considered in this study without compromising the performance of the structure.