When a signal beam propagates through the atmosphere, it is affected by turbulence. The Airy beam, with its characteristics of diffraction-free propagation, self-healing, and self-focusing, can effectively mitigate the adverse effects of atmospheric turbulence. In this paper, the coupling efficiency is constructed based on the theory of cross-spectral density, using a structure consisting of partially coherent Airy beams, a single lens, and a four-mode optical fiber. In addition, we also investigated the effects of atmospheric turbulence, lens parameters, and beam parameters on the coupling efficiency. The results indicate that under moderate turbulence conditions (scintillation index AR/AC = 5), the coupling efficiency of the partially coherent Airy beams is 25.7% higher than that of the Gaussian-Schell model beams.