We propose and analyze a finite element method for the Reynolds-Orr eigenvalue problem in wall-bounded shear-driven incompressible flows with arbitrary cross-section. This problem is similar to the Stokes eigenvalue problem but includes an additional term involving the strain rate tensor of the underlying laminar flow. This term renders one of the bilinear forms non-coercive, requiring adaptations to the standard spectral approximation framework for both the continuous and discrete eigenproblems. Using the theory of compact operators, we prove convergence for inf-sup stable finite elements and demonstrate that the proposed method provides accurate error estimates for both eigenvalues and eigenfunctions. We carry out various numerical tests to showcase how well the method performs and confirm our theoretical results' accuracy.