This paper studies the stabilization of Boolean control networks (BCNs) under aperiodic DoS attacks. DoS attacks are a common and destructive network attack that weaken the performance of a system by blocking the transmission of information between network nodes. We focus on analyzing aperiodic DoS attacks that obey an independent and identically distributed process. First, for aperiodic DoS attacks following an independent and identically distributed process, we obtain a necessary and sufficient condition for the system stabilization by transforming the attacked BCN into a probabilistic Boolean network. Then, the state feedback control is designed using the reachable set method to ensure the system stabilization. Finally, the effectiveness of the proposed theory and control method is illustrated by a simulation example.