In this work, the scattering of elastic plane waves by a rigid obstacle in a two-dimensional homogeneous isotropic elastic medium is studied. A stable node-based smoothed finite element method (SNS-FEM) and the transparent boundary condition (TBC) are proposed to solve this problem. Firstly, using Helmholtz decomposition，two scalar potential functions are introduced to divide the Navier equation into Helmholtz equations with the coupled boundary conditions for the elastic wave scattering problem, and the displacement of elastic wave equation is divided into compression wave and shear wave. Second, based on the analytical solutions of Helmholtz equations, the Dirichlet to Neumann (DtN) operators are deduced and truncated the open domain into a bounded domain. Then, based on the Taylor expansion, the gradient of the solution is expressed into a linear formulation through approximating the node-based smoothing region as a circle, which makes the solution of original node-based S-FEM (NS-FEM) stable. Lastly, the formulation of SNS-FEM with TBC is derived. Numerical examples show that SNS-FEM can obtain faster convergence, more stable and accurate solutions than standard FEM.