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Compared to the onshore wind turbines, the offshore floating wind turbines experience larger loads and displacements due to combined effects of the wind and waves. Suppressing the structural vibration of floating wind turbines with a passive tuned mass damper (TMD) device has the potential to be an effective and economical method. In this study, a barge-type floating wind turbine is chosen as the research object. The SIMPACK multi-body dynamics software is employed to establish the detailed model of the wind turbine. For achieving considerable load reduction performance, the spring stiffness, damping coefficient, and mass of the TMD system are optimized based on a simplified mathematical model. Furthermore, the genetic algorithm (GA) is used to achieve global optimal TMD parameters. To investigate the effect of the detailed drivetrain model on the wind turbine performances and comprehensively evaluate the structural vibration inhibition effect of the designed TMD, a coupled aero-hydro-servo-elastic model is constructed and simulated for different load cases. The time-domain and frequency-domain analysis of simulation results are performed. Moreover, a robustness study for TMD parameters obtained by the GA is also performed. Analysis results indicate that the flexibility and gear contact within the drivetrain has a noticeable effect on the wind turbine performances, and the designed TMD can significantly inhibit the structural loads and stabilize the electrical output power.