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A fundamental concern in progressing Airborne Wind Energy (AWE) operations towards commercial success, is guaranteeing that safety requirements placed on the systems are met. Due to the high dimensional complexity of AWE systems, however, formal mathematical robustness guarantees become difficult to compute. We draw on research from Hamilton-Jacobi (HJ) reachability analysis to compute the optimal control policy for tracking a flight path, while enforcing safety constraints on the system. In addition, the zero-sublevel set of the computed value function inherent in HJ reachability analysis indicates the backward reachable set, the set of states from which it is possible to safely drive the system into a target set within a given time without entering undesirable states. Furthermore, we derive a switching law, such that the safety controller can be used in conjunction with arbitrary least restrictive controllers to provide a safe hybrid control law. In such a setup, the safety controller is only activated when the system approaches the boundary of its maneuverability envelope. Such a hybrid control law is a notable improvement over existing robust control approaches that deteriorate performance by assuming the worst-case environmental and system behavior at all times. We illustrate our results via extensive simulation-based studies.