A novel framework for improving stability of virtual element method (VEM) by the graph neural networks (GNNs) is proposed. As a Galerkin method supporting arbitrary element discretization, VEM has significantly greatly advanced polygonal discretization methods. However, its absence of explicit shape functions has long posed challenges to numerical stability. To overcome this limitation, we propose GNE-VEM—a Graph Neural Networks Enhanced VEM (GNE-VEM) framework for optimizing its stability. GNE-VEM leverages a GNNs to adaptively determine the element-wise stabilization terms based on local mesh and geometric information, thereby reducing the dependence on heuristic parameter choices and mitigating artificial numerical errors. Specifically, by extracting the shape features of VEM elements via dimensionless quality metrics, GNE-VEM employs graph convolution for message aggregation and outputs a vector of stability coefficients. The evaluation results demonstrate that GNE-VEM generalizes robustly across mesh resolution, element shape arbitrariness, and loading and boundary conditions, achieving higher numerical accuracy than the standard VEM. Furthermore, when applied to other linear elasticity problems, GNE-VEM mitigates the shear locking and volumetric locking inherent in low-order VEM through accurate prediction of high-order strain energy, demonstrating its broad applicability to a range of two-dimensional solid mechanics problems.