In this paper, we develop an energy stable finite difference method for the problem of curve motion under anisotropic surface diffusion. The motion of curve by anisotropic surface diffusion is governed by the fourth-order (highly nonlinear) geometric evolution equation. As in Li and Bao (2021), we first split the fourth-order evolution equation into two second-order equations, where the position vector of curve and the weighted curvature are treated as unknowns. Instead of using the arclength parameter, we introduce a Lagrangian coordinate parameter such that a closed curve can be parametrized over a fixed interval so that the equations can be represented using the new parameter. We then propose a linearly semi-implicit finite difference method to discretize these two equations, and prove that the scheme satisfies discrete energy dissipation so it is energy stable under suitable condition on the anisotropic surface energy. To show the applicability of our present method, we perform several numerical tests on various initial curves and different anisotropic energies. The numerical results show that our scheme is indeed energy dissipative and conserves even the total area well.