Based on the extended Huygens-Fresnel principle, the expression for the spectral degree of coherence of the partially coherent Hermite-Gaussian (H-G) beams propagating through atmospheric turbulence is derived by using the quadratic approximation of Rytov's phase structure function, and the influence of atmospheric turbulence on the spatial correlation is studied. It is shown that the oscillatory behavior and phase singularities of the degree of spectral coherence may appear when partially coherent H-G beams propagate through atmospheric turbulence, which is very different from the behavior of Gaussian-Schell-model beams. But, the oscillatory behavior becomes weaker with increasing turbulence, and even disappears when the turbulence is strong enough. The smaller the coherence parameter, the less the spatial correlation property of partially coherent H-G beams is affected by the turbulence. In addition, a comparison between the mean-squared width of the spectral degree of coherence and the mean-squared width of the spectral intensity is also given. Some interesting results are obtained, and are explained with regard to their physical nature.