Safe and socially compliant navigation in a crowded environment is essential for social robots. Numerous research efforts have shown the advantages of deep reinforcement learning techniques in training efficient policies, while most of them ignore fast-moving pedestrians in the crowd. In this paper, we present a novel design of safety measure, named Risk-Area, considering collision theory and motion characteristics of different robots and humans. The geometry of Risk-Area is formed based on the real-time relative positions and velocities of the agents in the environment. Our approach perceives risk in the environment and encourages the robot to take safe and socially compliant navigation behaviors. The proposed method is verified with three existing well-known deep reinforcement learning models in densely populated environments. Experiment results demonstrate that our approach combined with the reinforcement learning techniques can efficiently perceive risk in the environment and navigate the robot with high safety in the crowds with fast-moving pedestrians.