A hybrid cavity-magnon system is proposed to enhance the steady-state entanglement and quantum steering between magnons by utilizing the Kerr nonlinearity of magnons and cross-Kerr nonlinearity between microwave cavities. The system consists of two microwave cavities and two yttrium iron garnet (YIG) spheres. By tuning the frequency detuning between microcavities and magnons, the cavity-magnon coupling coefficient, and the cross-Kerr nonlinearity coefficient, this scheme enhances quantum entanglement and steering between magnons. Furthermore, the constructed system exhibits strong robustness against environment-induced decoherence.