We show that multiplexing (Here, the term "multiplexing" means a special network topology where a one-layer network is connected to another one-layer networks through coupling between replica nodes. In the present paper, this term does not refer to the signal processing issues and telecommunications.) allows us to control noise-induced dynamics of multilayer networks in the regime of stochastic resonance. We illustrate this effect on an example of two- and multi-layer networks of bistable overdamped oscillators. In particular, we demonstrate that multiplexing suppresses the effect of stochastic resonance if the periodic forcing is present in only one layer. In contrast, multiplexing allows us to enhance the stochastic resonance if the periodic forcing and noise are present in all the interacting layers. In such a case, the impact of multiplexing has a resonant character: the most pronounced effect of stochastic resonance is achieved for an appropriate intermediate value of coupling strength between the layers. Moreover, multiplexing-induced enhancement of the stochastic resonance can become more pronounced for the increasing number of coupled layers. To visualize the revealed phenomena, we use the evolution of the dependence of the signal-to-noise ratio on the noise intensity for varying strength of coupling between the layers.