This paper studies the design of a state-feedback delay-dependent H for vibration attenuation problem of a seismic-excited container crane subject to having time-varying actuator delay, L-2 type disturbances and actuator saturation. First, a sufficient delay-dependent stability criterion is developed by choosing a Lyapunov-Krasovslcii functional candidate based on matrix inequalities for a stabilizing H-infinity synthesis. To convexify the Bilinear Matrix Inequality (BMI) based optimization problem involved in the delay dependent conditions; a cone complementary linearization method is adopted to find a sub-optimal solution. The proposed method also utilizes convex description of nonlinear saturation phenomenon by means of convex hull of some linear feedback which leads to a few additional ellipsoidal conditions in terms of Linear Matrix Inequalities (LMIs). By use of the proposed method, a suboptimal controller with maximum allowable delay bound and minimum allowable disturbance attenuation level can be easily obtained by a convex optimization technique. In order to show effectiveness of the proposed approach, a five Degrees-of-Freedom (DOF) container crane structure is modeled using a spring-mass-damper subsystem. The system is then simulated against the real ground motions of Kobe and Northridge earthquakes. Finally, the time history of the crane parts displacements, accelerations, control forces and frequency responses of the both uncontrolled and controlled cases are presented. Additionally, the performance of the proposed controller is also compared with a nominal state-feedback H-infinity controller performance. Simulation results show that, in spite of the actuator saturation, the designed controller is all effective in reducing vibration amplitudes of crane parts and guarantees stability at maximum actuator delay.