Focusing on the safe operation of rail transit during earthquakes, the finite element method is used to construct a wheel-track-subgrade dynamics model in this study. Through spring-damper units, the relationship between the rail and the track slab and the connection between the track slab and the subgrade are established. A method for establishing a viscoelastic artificial boundary is proposed. Four seismic waves—the Tianjin wave, the El Centro wave, the Taft wave, and the Qian’an wave—are selected as the seismic input waveforms, and only the impact of the lateral ground motion on the wheel-track-subgrade system is considered. In this paper, the ground motion problem is transformed into a wave source problem, the seismic input is transformed into an equivalent load acting on the artificial boundary, and the wave input of the viscoelastic artificial boundary is realized. The normalization method is used to process the seismic waves, and a method that converts the input of the seismic waves into equivalent loads is proposed. The changing laws of different dynamic response indexes under the influence of the four waveforms are studied. Under the action of the Tianjin wave, the wheel-rail dynamic response is very violent near the acceleration peak, whereas, after the peak, all dynamic response indexes are within a safe range. Under the effect of the El Centro wave, the collision between the wheel and the track is relatively violent, and the train is already in a dangerous state. Under the action of the Taft wave, due to the sudden action of the peak ground motion acceleration, the displacement between the wheel and the track increases instantaneously, causing the train to derail. Under the action of the Qian’an wave, the force between the wheel and rail changes approximately linearly with respect to the frequency of the ground motion, and all dynamic response indexes are within a safe range. The vibration intensity of the four seismic waves is amplified by an intensity expansion factor. Except for the Tianjin wave, the amplified seismic wave has a greater impact on safe train operations. This paper can provide a reference for research on the running safety of trains under similar dynamic disturbance conditions.