It is demonstrated theoretically that a thin layer of an anisotropic antiferromagnetic (AFM) insulator can effectively conduct spin current by excitation of a pair of evanescent AFM spin wave modes. The spin current flowing through the AFM is not conserved due to the interaction between the excited AFM modes and the AFM lattice, and, depending on the excitation conditions, can be either attenuated or enhanced. When the phase difference between the excited evanescent modes is close to π/2, there is an optimum AFM thickness for which the output spin current reaches a maximum that can significantly exceed the magnitude of the input spin current. The spin current transfer through the AFM depends on the ambient temperature and increases substantially when temperature approaches the Néel temperature of the AFM layer.