This article considers asymptotic stability and stabilization of Markovian jump Boolean networks (MJBNs) with stochastic state-dependent perturbation. By defining an augmented random variable as the product of the canonical form of switching signal and state variable, asymptotic stability of an MJBN with perturbation is converted into the set stability of a Markov chain (MC). Then, the concept of induced equations is proposed for an MC, and the corresponding criterion is subsequently derived for asymptotic set stability of an MC by utilizing the solutions of induced equations. This criterion can be, respectively, examined by either a linear programming algorithm or a graphical algorithm. With regards to the stabilization of MJBNs, the time complexity is reduced to a certain extent. Furthermore, all time-optimal signal-based state feedback controllers are designed to stabilize an MJBN towards a given target state. Finally, the feasibility of the obtained results is demonstrated by two illustrative biological examples.