Symplecticity is a significant property of stochastic Hamiltonian systems, and the symplectic methods are very attractive. Compared with the classical non-exponential Runge-Kutta methods, the exponential Runge-Kutta methods are more suitable for stiff problems. Therefore, the focus of this paper is on constructing stochastic symplectic exponential Runge-Kutta (SSERK) integrators for semilinear stochastic differential equations (SDEs) driven by multiplicative noise. The first is to establish the symplectic conditions of stochastic exponential Runge-Kutta (SERK) methods. It can be found that when the stiffness matrix is 0, these conditions will degenerate into the symplectic conditions of classical stochastic Runge-Kutta methods. Based on this idea, we construct a class of SSERK integrators with remarkable properties of structure-preservation. In addition, we verify the existence of the quadratic first integral of the stochastic Hamiltonian system and investigate the connection between preserving both the quadratic first integral and the symplectic structure by the SERK methods. Numerical experiments demonstrate a better structure-preserving ability and a higher accuracy of the SSERK integrators in solving the considered semilinear SDEs than the corresponding stochastic symplectic Runge-Kutta integrators. Excitingly, the SSERK integrators perform well when applied to the temporal discretization of stochastic nonlinear Schrödinger equation.