To achieve lightweight vibration control in offshore wind turbines (OWTs), this study proposes the utilization of two types of tuned inerter-based dampers (TIBDs), namely, the tuned viscous mass damper (TVMD) and tuned inerter damper (TID), to mitigate the coupled responses of an OWT induced by wind and wave loads. First, a multi-degree-of-freedom (MDOF) lumped mass model is established to represent an OWT equipped with a TIBD. Then, a novel structural control module for TIBDs is developed and seamlessly integrated into the modular numerical analysis code OpenFAST. This enables fully coupled aero-hydro-servo-elastic simulations of the OWT-TIBD system. Using the established MDOF model, a TIBD parameter optimization design method is proposed to minimize the peak of the non-dimensional displacement frequency response function. Additionally, a parametric study is performed to investigate the effect of inertance and span height on the control performance. Finally, the performance of the TIBD in attenuating both fatigue and extreme loads is evaluated through nonlinear fully coupled time-domain simulations using OpenFAST and compared to traditional tuned mass dampers (TMDs). The results showed that a lightweight and small damper displacement TIBD is capable of effectively dampening OWT vibrations, yielding a better mitigation effect than that provided by a bulky TMD. Overall, the reduced installation requirements and the commendable control efficiency of a TIBD highlight its potential as a promising candidate for passive control devices in OWT applications.