To accommodate the stringent requirements of enhanced coverage quality and improved spectral efficiency, simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS)-aided communication has been perceived as an interesting research topic. This paper investigates a downlink STAR-RIS-aided non-orthogonal multiple access (NOMA) system, where a STAR-RIS is deployed to enhance the transmission qualities between users and a multiple-antenna base station (BS). The considered STAR-RIS utilizes the energy splitting (ES) protocol to serve a pair of NOMA users located at both sides of STAR-RIS. Based on the ES protocol, each reconfigurable element can operate in the modes of transmission and reflection simultaneously. In an effort to characterize the secrecy performance, we first derive the closed-form expressions of secrecy outage probability (SOP) for STAR-RIS-aided NOMA system. Then, the asymptotic performance of the derived SOP is analyzed. For gleaning further insights, secrecy diversity order (SDO) is derived according to the asymptotic approximation in the high signal-to-noise ratio and main-to-eavesdropper ratio regimes. As a further advance, the system parameters are optimized to minimize the SOP of the system. Our analytical results demonstrate that the multiple-antenna BS has almost no impact on SDO for STAR-RIS-aided NOMA system. In simulations, it is demonstrated that the theoretical results match with the simulation results very well and the SOP of STAR-RIS-aided NOMA is less than that of conventional orthogonal multiple access (OMA) system obviously.