Offshore cranes often work in harsh sea conditions. As a result, they may suffer from various external disturbances, especially the persistent and unpredictable ship motion caused by sea waves, which will introduce severe disturbances to the crane dynamics, including the unactuated part. Due to this reason, the control problem for offshore boom cranes presents great challenges, for which only boundedness or stability, instead of the generally desired asymptotic stability, of the closed-loop system can be usually guaranteed by currently available methods. For this problem, a novel nonlinear control strategy for an offshore crane is proposed in this brief, which ensures that the closed-loop system's equilibrium point is asymptotically stable even in the presence of the persistent ship roll disturbances. The proposed control law is further extended to be an output feedback controller to deal with the situations when the velocities are unavailable, still guaranteeing asymptotic stability. Finally, some experimental results are provided to validate the efficiency of the proposed controllers.