Deep reinforcement learning (DRL) has a great potential for solving complex decision-making problems in autonomous driving, especially in mixed-traffic scenarios where autonomous vehicles and human-driven vehicles (HDVs) drive together. Safety is a key during both the learning and deploying reinforcement learning (RL) algorithms process. In this paper, we formulate the on-ramp merging as a Markov Decision Process (MDP) problem and solve it with an off-policy RL algorithm, i.e., Soft Actor-Critic for Discrete Action Settings (SAC-Discrete). In addition, a motion predictive safety controller including a motion predictor and an action substitution module, is proposed to ensure driving safety during both training and testing. The motion predictor estimates the trajectories of the ego vehicle and surrounding vehicles from kinematic models, and predicts potential collisions. The action substitution module updates the actions based on safety distance and replaces risky actions, before sending them to the low-level controller. We train, evaluate and test our approach on a gym-like highway simulation with three different levels of traffic modes. The simulation results show that even in harder traffic densities, our proposed method still significantly reduces collision rate while maintaining high efficiency, outperforming several state-of-the-art baselines in the considered on-ramp merging scenarios. The video demo of the evaluation process can be found at: https://www.youtube.com/watch?v=7FvjbAM4oFw