Underwater visual localization is an essential technique for the autonomous operation of underwater robots. However, the unique underwater image characteristics, including refraction, sparse features, and severe noise, pose an enormous challenge to it. For addressing these issues, this paper proposes an open-source fiducial-based underwater stereo visual-inertial localization method under the extended Kalman filter (EKF) framework, which is called FBUS-EKF. First, the refraction is corrected by the refractive camera model and akin triangulation. Second, the fiducial marker and a novel marker pose estimation method are applied to alleviate the adverse effect of sparse features. Third, the EKF is utilized to fuse the inertial and visual information so as to reject the serious noise. Finally, extensive experiments on a test bench demonstrate the effectiveness of the FBUS-EKF method, where the typical localization error is less than 3%, namely, the average error is lower than 3 cm within one meter. The obtained results reveal that the FBUS-EKF method has the prospect to be applied in the precise short-range operation and the localization for underwater robots, which offers a valuable insight for further autonomous underwater task.