Reinforcement learning (RL) has been widely used in decision-making and control tasks, but the risk is very high for the agent in the training process due to the requirements of interaction with the environment, which seriously limits its industrial applications such as autonomous driving systems. Safe RL methods are developed to handle this issue by constraining the expected safety violation costs as a training objective, but the occurring probability of an unsafe state is still high, which is unacceptable in autonomous driving tasks. Moreover, these methods are difficult to achieve a balance between the cost and return expectations, which leads to learning performance degradation for the algorithms. In this paper, we propose a novel algorithm based on the long and short-term constraints (LSTC) for safe RL. The short-term constraint aims to enhance the short-term state safety that the vehicle explores, while the long-term constraint enhances the overall safety of the vehicle throughout the decision-making process, both of which are jointly used to enhance the vehicle safety in the training process. In addition, we develop a safe RL method with dual-constraint optimization based on the Lagrange multiplier to optimize the training process for end-to-end autonomous driving. Comprehensive experiments were conducted on the MetaDrive simulator. Experimental results demonstrate that the proposed method achieves higher safety in continuous state and action tasks, and exhibits higher exploration performance in long-distance decision-making tasks compared with state-of-the-art methods.