To demonstrate the operational safety of the equatorial space elevator system and the mechanism of the influence of the climber moving on the system vibration characteristics, the stability of the system and the effect of the climber moving with different acceleration time and deceleration time on the system residual oscillation were studied. A rigid tether model with two-DOF was established by Lagrange method. The stability of the topological equivalent linearization system based on the small oscillation angle assumption was analyzed at the equilibrium point. Two parameters which are the time ratio of acceleration and deceleration phases with the cruise phase were introduced to study the response of the system. Results show that the system had gradual stability characteristics at the equilibrium point both inside and outside the equator plane under the condition of atmospheric damping. The motion of the climber would cause the tether to oscillate in the equatorial plane, which was mainly caused by the Coriolis force. There was always a minimum amplitude of system residual oscillation in-plane for the deceleration time ratio, which is related to the acceleration time ratio. The amplitude of system residual oscillation could be controlled in level of 10^-3degree by optimizing the deceleration time ratio, and the simulation result indicated that increasing the cruise speed and decreasing the acceleration time of the climber could shorten the time for the climber to run to the target track. Optimizing the time ratio of acceleration and deceleration phases could suppress the residual oscillation and improve the efficiency and economy of the system.