Racing demands each vehicle to drive at its physical limits, when any safety infraction could lead to catastrophic failure. In this work, we study the problem of safe reinforcement learning (RL) for autonomous racing, using the vehicle's ego-camera view and speed as input. Given the nature of the task, autonomous agents need to be able to 1) identify and avoid unsafe scenarios under the complex vehicle dynamics, and 2) make sub-second decision in a fast-changing environment. To satisfy these criteria, we propose to incorporate Hamilton-Jacobi (HJ) reachability theory, a safety verification method for general non-linear systems, into the constrained Markov decision process (CMDP) framework. HJ reachability not only provides a control-theoretic approach to learn about safety, but also enables low-latency safety verification. Though HJ reachability is traditionally not scalable to high-dimensional systems, we demonstrate that with neural approximation, the HJ safety value can be learned directly on vision context -- the highest-dimensional problem studied via the method, to-date. We evaluate our method on several benchmark tasks, including Safety Gym and Learn-to-Race (L2R), a recently-released high-fidelity autonomous racing environment. Our approach has significantly fewer constraint violations in comparison to other constrained RL baselines in Safety Gym, and achieves the new state-of-the-art results on the L2R benchmark task. We provide additional visualization of agent behavior at the following anonymized paper website: https://sites.google.com/view/safeautonomousracing/home