In this article, we propose a novel framework for reliability-based co-design of stochastic dynamical systems. Specifically, it solves a combined reliability-based design optimization problem of the plant design with parametric uncertainties and the state-constrained stochastic optimal control. In the past, co-design strategies have been applied primarily in a deterministic manner, often employing open-loop optimal control techniques in generating solutions and system design insights. Here we extend the scope of co-design methods to a specific class of stochastic dynamical systems by providing a stationary solution to its Hamiltonian-Jacobi-Bellman equation, based on logarithmic transformation and linearization. For reliability assessment, Monte Carlo Simulation, Performance Measurement Approach, and Sequential Optimization and Reliability Assessment methods are evaluated with respect to accuracy and computational expense. The proposed method is demonstrated and assessed using both an analytical example and a co-design problem involving an actively controlled spring-mass-damper system.