This paper has a twofold aim: to assess the performance of tuned-inerter dampers (TIDs) to reduce vibrations in cable structures subjected to seismic ground motions; and to extend an existing TID-design method for cables to a practical scenario of multiple cables in a cable-stayed bridge. In this study, TIDs are installed between the cables and the bridge’s deck, the cables being excited at both their ends by the response of the bridge’s deck and pylon to seismic ground motions. The seismic hazard is described in a general manner by rates of earthquakes and synthetic ground-motion time histories with respect to their moment magnitudes and epicentral distances. Two approaches that use an existing deterministic method to design TIDs to reduce vibrations in cables are discussed. The first one, perhaps more effective but impractical, assumes that each cable’s response is reduced by its own independently-designed TID. The second approach, more realistic and practical, proposes the design of one single TID for all cables to reduce their mid-span response in an optimal way. Numerical results are shown for cable models in the Evripos Bridge in Greece, for which a unique TID is designed for all cables in the bridge, subjected to a hypothetical seismic-hazard scenario. It is shown that the TID-controlled cables have generally a better response, that is a lower average maximum absolute mid-span displacement for most of the cables, and lower variability in the response.