The berthing of maritime autonomous surface ships (MASSs) is a challenging operation even for an experienced captain due to the required complicated maneuvering at low speeds in a constrained water space. This paper addresses safety-aware automatic berthing of MASSs moving in a constrained water region with stationary and moving obstacles. A safety-critical constrained anti-disturbance control method is proposed for the automatic berthing of MASSs subject to velocity constraints, input constraints, and collision-avoidance constraints, as well as ocean disturbances. At the kinematic level, desired linear velocities and yaw rate are prescribed by incorporating a line-of-sight guidance scheme to achieve position-heading stabilization. At the kinetic level, an anti-disturbance kinetic control law is designed based on an extended state observer. Through converting the input constraints to velocity constraints on guidance signals, the safety-guidance signals are obtained by solving a quadratic programming problem subject to the state, input, and collision-avoidance constraints. It is proven that an MASS controlled by using the proposed method is input-to-state safe in the presence of ocean disturbances. Simulation results are elaborated to substantiate the efficacy of the proposed safety-certified control law for the automatic berthing of an MASS subject to physical constraints and environmental disturbances.