The observations on galaxy rotation curves show significant discrepancies from the Newtonian theory. This issue could be explained by the effect of the anisotropy of the spacetime. Conversely, the spacetime anisotropy could also be constrained by the galaxy rotation curves. Finsler geometry is a kind of intrinsically anisotropic geometry. In this paper, we study the effect of the spacetime anisotropy at galactic scales in the Finsler spacetime. It is found that the Finslerian model has close relations with the Milgrom's MOND. By performing the best-fit procedure to the galaxy rotation curves, we find that the anisotropic effects of the spacetime become significant when the Newtonian acceleration GM/r(2) is smaller than the critical acceleration a(0). Interestingly, the critical acceleration a0, although varies between different galaxies, is in the order of magnitude cH(0)/2 pi similar to 10(-10) ms(-2).