Bioinspired robotic fish have garnered significant interest in recent decades due to their agile maneuverability and diverse marine applications. However, accurate modeling and physical prototyping of flexible fan-shaped membranous pectoral fins remain challenging. This paper presents the mathematical framework, modular design, and fabrication of an underwater robotic fish propelled by a pair of single-degree-of-freedom (DOF) membranous fins. To capture the complex fluid-structure interactions, the Morison force model is applied to establish a dynamic formulation that couples the fin motion, the robot body, and the surrounding fluid. A crank-rocker mechanism is proposed to drive the fin rays, generating fan-shaped membrane undulations for backward fluid propulsion while allowing straightforward phase shift adjustments. Experimental results demonstrate that the fabricated prototype achieves a peak forward swimming speed of approximately 0.59 body lengths per second (BL/s) and a turning radius of 0.14–0.53 BL. Furthermore, the modular architecture accommodates a water-quality monitoring payload, thereby demonstrating its capability for in situ measurement of total dissolved solids (TDS) and turbidity. This work provides a validated computational and experimental framework for advanced control and environmental monitoring in aquatic applications.