Ensuring safe real-time control of ship-mounted cranes in unstructured transportation environments requires handling multiple safety constraints while maintaining effective payload transfer performance. Unlike traditional crane systems, ship-mounted cranes are consistently subjected to significant external disturbances affecting underactuated crane dynamics due to the ship's dynamic motion response to harsh sea conditions, which can lead to robustness issues. To tackle these challenges, we propose a robust and safe model predictive control (MPC) framework and demonstrate it on a 5-DOF crane system, where a Stewart platform simulates the external disturbances that ocean surface motions would have on the supporting ship. The crane payload transfer operation must avoid obstacles and accurately place the payload within a designated target area. We use a robust zero-order control barrier function (R-ZOCBF)-based safety constraint in the nonlinear MPC to ensure safe payload positioning, while time-varying bounding boxes are utilized for collision avoidance. We introduce a new optimization-based online robustness parameter adaptation scheme to reduce the conservativeness of R-ZOCBFs. Experimental trials on a crane prototype demonstrate the overall performance of our safe control approach under significant perturbing motions of the crane base. While our focus is on crane-facilitated transfer, the methods more generally apply to safe robotically-assisted parts mating and parts insertion.