Most real-world complex systems consist of multiple subsystems that interact with each other, which can be represented as multilayer networks. Implications of such multilayer topologies have been studied for system robustness, cascading failures, reaction-diffusion dynamics, etc. Meanwhile, limited attention has been paid to the effects of inter-layer coupling on the dynamical stability of multilayer networks. Here we study the stability of multilayer networks and its relationships with the stabilities of networks within individual layers. Specifically, we generated two random networks following the method of May's random ecological network model, and then coupled them to create a random multilayer network. Two parameters were systematically varied: the strength of within-layer connections, and the strength of inter-layer coupling. Stabilities of those networks were evaluated by calculating eigenvalues of their weighted adjacency matrices. We found that the whole multilayer network became destabilized by sufficiently strong inter-layer coupling even if within-layer connections were adjusted so that individual networks were stable. Also, the network instability induced by inter-layer coupling was manifested as the network size of individual networks increases. Our study illustrates that the strength of inter-layer coupling plays a critical role in determining the stability of the entire system.