An optimized static gait that combines pose optimization, motion sequence optimization, and a novel high-level planning algorithm is proposed for quadruped robots to walk on stairs. Firstly, an optimized pose is determined for the robot to stand on stairs statically. Then, a climbing gait cycle with an optimized motion sequence is presented, which takes the robot from one position and pose to another position and pose. Finally, a high-level planning algorithm is proposed to adjust the step length in each gait cycle to enable the robot to safely walk along the stairs. The proposed static gait maximizes the stair-climbing capability significantly while still guaranteeing walking safety, which provides a general solution for quadruped robots to walk on stairs of different sizes. Several simulations in V-REP are presented to evaluate the effectiveness of the optimized static gait generation technique in improving the stair-climbing capability. Compared to other quadruped robots developed recently, the robot tested in this paper can walk on a regular staircase with a rise of 20 cm and an inclination of 37.6°, and can also climb over a few steep narrow steps with a rise of 18 cm and a run of 5 cm.