We consider transverse linear stability of line solitary waves for the 2-dimensional Benney-Luke equation which arises in the study of small amplitude long water waves in 3D. In the case where the surface tension is weak or negligible, we find a curve of resonant continuous eigenvalues of the linearized operator in a neighborhood of $\lambda=0$. Time evolution of these resonant continuous eigenmodes is described by a 1D damped wave equation in the transverse variable. In exponentially weighted space whose weight function increases in the direction of the motion of the line solitary wave, the other part of solutions to the linearized equation decays exponentially as $t\to\infty$.