In the information age, achieving safe control for complex networked systems presents a formidable challenge. This paper investigates the safe control problem for a class of state-triggered impulsive Boolean control networks (SIBCNs) with input constraints. First, based on the hybrid index model and the semi-tensor product (STP), a mathematical model with state-dependent input constraints is formulated and equivalently converted into an unconstrained impulsive Boolean control network via input transformation. Second, the concept of forward completeness is added to avoid the Zeno phenomenon, and necessary and sufficient conditions for the solvability of the safe control problem are established within the quotient system. Furthermore, a controller based on admissible paths and state space partitioning is designed, and the admissible feedback under input constraints is constructed. Finally, a state feedback controller design algorithm is proposed to address the safe control problem in the time domain. Simulation results demonstrate that the proposed algorithm effectively solves the safe control problem while satisfying the prescribed input constraints.