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Offshore wind offers a compelling pathway for large-scale green hydrogen production, but its variability creates a difficult design problem: electrolyzers must be large enough to capture high-wind periods without becoming underutilized, while aggressive operation can accelerate stack degradation and undersizing can leave substantial wind energy curtailed. Recent studies confirm that degradation and wind variability can materially alter hydrogen economics, making lifetime-aware sizing and operation important for credible offshore projects. This study develops a multi-objective optimization framework for the integrated sizing and energy management of an offshore wind-powered green hydrogen plant, explicitly resolving the competing objectives of minimizing levelized cost of hydrogen (LCOH), cumulative electrolyzer degradation, and wind-energy curtailment. Hourly offshore wind generation is coupled with a nonlinear electrolyzer model incorporating part-load efficiency, minimum loading, start-up/shutdown behavior, operating-state-dependent degradation, hydrogen production, and stack replacement. Wind-farm capacity, electrolyzer rating, hydrogen-storage capacity, and operational setpoints are optimized using a multi-objective evolutionary algorithm to construct a stability-feasible Pareto front rather than collapsing conflicting objectives into a predetermined weighted cost function. Simulation results identify a minimum-cost configuration with an LCOH of approximately $3.84/kg H₂, but with 16.7% wind curtailment and comparatively aggressive electrolyzer operation. Prioritizing asset lifetime reduces cumulative degradation by approximately 31.8%, although LCOH increases to $4.23/kg H₂. Conversely, the minimum-curtailment configuration reduces curtailed wind energy to approximately 6.4%, but requires greater electrolyzer and storage capacity. The selected balanced Pareto solution achieves an LCOH of approximately $3.98/kg H₂, while reducing electrolyzer degradation by 23.6% and wind curtailment by 38.9% relative to the minimum-cost configuration. The results show that modest movement away from the purely cost-optimal design can capture a substantial share of the available durability and renewable-utilization benefits before diminishing returns emerge. Sensitivity analyses further indicate that wind-resource quality, electrolyzer CAPEX, stack-replacement cost, degradation severity, and hydrogen-storage cost can shift the preferred design, reinforcing the need for site-specific optimization. The study contributes a lifetime-aware framework that links offshore wind utilization, electrolyzer aging, and hydrogen economics within one transparent decision space. Practically, the resulting Pareto maps can help developers avoid both costly oversizing and short-sighted cost minimization while supporting more efficient use of offshore renewable resources. The principal limitation is the simulation-based representation of long-term degradation and wind uncertainty; consequently, future work should incorporate multi-year offshore measurements, probabilistic resource forecasts, detailed balance-of-plant dynamics, and experimental electrolyzer-aging validation. Overall, the findings demonstrate that economically competitive offshore hydrogen development depends not simply on maximizing production, but on balancing hydrogen cost, equipment lifetime, and otherwise-curtailed wind energy over the full project life.