Modeling dynamic urban streets is a critical challenge in the field of autonomous driving. With the evolution of 3D Gaussian Splatting (3DGS) technique, the efficient photo-realistic view synthesis can be achieved with high quality. However, existing dynamic scene modeling methods reconstruct the complex background of driving scenarios with unexpected blurries and color noises. To address this gap, we propose Street Depth-Aware Gaussian, dubbed as SeeDepthGaussian, which introduces a depth regularization method by utilizing monocular depth into open autonomous driving scenarios. This enables the recovery of more accurate scene details under depth supervision. Furthermore, the incorporation of Global-Local Depth Normalization further refines depth regularization by better capturing depth variations at different scales. Extensive experiments on the Waymo dataset indicate that SeeDepthGaussian achieves superior rendering quality compared to mainstream advanced 3DGS-based methods, and requires fewer computational resources as well as less training time than NeRF-based methods.