In this article, two boundary feedback controllers are designed via the backstepping approach for a class of gantry crane systems. To provide an accurate and concise representation of the dynamic behavior, the gantry crane with a flexible cable is described by a hybrid system. The hybrid system is formed by an ordinary differential equation coupled with a partial differential equation. In the first control strategy, a backstepping-based boundary state-feedback controller is proposed for the gantry crane to transport a payload to an expected position with less shaking. In the second control strategy, a boundary output-feedback controller is explored with an observer estimating the inaccessible states. By using the backstepping technique and kernel functions, the original systems with different control strategies are transformed into target systems. By using the operator semigroup and Lyapunov stability theories, the target system is proven to be well-posed and exponentially stable, respectively. Finally, numerical simulations and comparisons are provided to illustrate the efficiency and the advantages of the proposed methods.