The dynamic behavior of stay cables has a significant impact on the safety and serviceability of cable-stayed bridges. As tuning such dynamic behavior could be effectively achieved by a damping increase on stay cables, this paper investigates on the feasibility of increasing damping on two stay cables simultaneously through interconnecting them with a negative stiffness damper (NSD). It presents the passive realization of the NSD through the following process. First, under harmonic excitations, the steady-state dynamic responses of the two cables in the network are derived. Then, the asymptotic solutions for the additional modal damping ratios are formulated with the critical viscous damping and negative stiffness determined approximately. Subsequently, a parametric analysis is performed to verify the theoretical derivations using two stay cables of a real long-span cable-stayed bridge, under a series of numerical evaluations consisting of sinusoidal excitations and white noises vibrational responses for both cables. Both the theoretical and numerical results show superior damping enhancement by the NSD, in that the vibration responses of the two cables are reduced remarkably.