Marine cranes play an important role in many hoisting scenes. However, due to the ship motion excitation and crane operation, the payload swing is unavoidable, hence putting the operators in danger. Thus, this paper proposes two active damping methods to suppress the payload swing by a 3‐DOF cable‐driven parallel robot (CDPR). First, the structure of the 3‐DOF CDPR is briefly introduced. Then, the dynamic model of the system is modeled as a constrained spherical pendulum with moving base excitations. Meanwhile, inspired by the air damping phenomenon, two active damping methods are presented to damp the payload swing naturally and smoothly. Furthermore, the payload swing under ship motion excitation and external hitting disturbances is analyzed, the influence mechanism of system damping parameters on the payload anti‐swing is discussed, and the robustness of the proposed damping methods is verified. By theoretical analysis and numerical simulations, some interesting discoveries are revealed: (1) the proposed damping methods can effectively suppress the big offset of payload swing angle inducing by external hitting disturbance, but the system might reach a dynamic equilibrium with a tilting hoist cable; (2) the phenomenon of “off‐center spherical pendulation” might be induced by improper setting of the damping parameters.