Recently, binaural audio synthesis (BAS) has emerged as a promising research field for its applications in augmented and virtual realities. Binaural audio helps users orient themselves and establish immersion by providing the brain with interaural time differences reflecting spatial information. However, existing BAS methods are limited in terms of phase estimation, which is crucial for spatial hearing. In this paper, we propose the \textbf{DopplerBAS} method to explicitly address the Doppler effect of the moving sound source. Specifically, we calculate the radial relative velocity of the moving speaker in spherical coordinates, which further guides the synthesis of binaural audio. This simple method introduces no additional hyper-parameters and does not modify the loss functions, and is plug-and-play: it scales well to different types of backbones. DopperBAS distinctly improves the representative WarpNet and BinauralGrad backbones in the phase error metric and reaches a new state of the art (SOTA): 0.780 (versus the current SOTA 0.807). Experiments and ablation studies demonstrate the effectiveness of our method.