An unmanned aerial vehicle (UAV) is employed to sequentially visit the specific waypoints and provide offloading services for nearby devices. Most of the current works optimized the UAV-enabled offloading according to a single criterion while neglecting necessary optimizations and constraints for flight safety of the UAV. This motivates us to study the optimization problem of the UAV from a multi-objective viewpoint by considering the UAV's flight safety. A constrained multi-objective optimization problem (CMOP) involving two objective functions about the energy-efficient offloading and safe path planning is formulated for the UAV. To solve the formulated CMOP, we present a constrained decomposition-based multi-objective evolution algorithm. To further improve the algorithm, we particularly utilize the infeasible individuals with great objective values, which provide useful information for improving the optimized objective values during the evolution process. Finally, experimental results demonstrate that compared with the existing works, our scheme is beneficial to simultaneously reduce energy consumption and ensure safe flight for the UAV.