Realizing more complex functions with more simple structure, lower weight/price and easier control system is still a great challenge for prosthetic hand. This paper proposed a novel underactuated prosthetic hand with flexible multi-joint finger structure based on Electroencephalogram (EEG) control. It contains 5 fingers moving independently and driven by a linear actuator each. Each finger is designed as similar integrated structure using parametric modeling method and is 3D printable, which can realize rapid fabrication and customization for different anthropometric dimensions of amputees. The adjacent phalanges are connected with flexible joints, and an elastic steel sheet is used to drive the joint to rotate. Except that the thumb contains 2 joints, each of the other four fingers contains 3 joints. We constructed a portable prototype and developed an EEG-based control system for the prosthetic hand. The pilot experiment shows it can achieve basic movements like pointing, pinching and holding, which verifies the effectiveness of this design.