Quantum steering is a kind of quantum correlations stronger than entanglement but weaker than Bell nonlocality. In an optomechanical system pumped by squeezed light and driven in the red sideband, we study—under thermal effects—stationary Gaussian steering and its asymmetry of two mechanical modes. In the resolved sideband regime using experimentally feasible parameters, we show that Gaussian steering can be created by quantum fluctuations transfer from the squeezed light to the two mechanical modes. Moreover, one-way steering can be observed by controlling the squeezing degree or the environmental temperature. A comparative study between Gaussian steering and Gaussian Rényi-2 entanglement of the two considered modes shows on one hand that both steering and entanglement suffer from a sudden death-like phenomenon with early vanishing of steering in various circumstances. On the other hand, steering is found stronger than entanglement, however, remains constantly upper bounded by Gaussian Rényi-2 entanglement, and decays rapidly to zero under thermal noise.