Combination of simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) and non-orthogonal multiple access (NOMA) is a win-win strategy which can significantly enhance the coverage performance. However, eavesdroppers may enjoy similar performance gains as the legitimate users. To solve this problem, an artificial noise (AN) assisted secure communication strategy is proposed to maximize the secrecy rate. An alternating optimization (AO) based iterative algorithm leveraging the classical successive convex approximation (SCA) and the semidefinite relaxation (SDR) techniques is proposed to derive the optimal AN model and the RIS parameters. It is found that the proposed algorithm provides better secrecy performance with less AN power compared with the benchmark schemes. More RIS elements help reducing the AN power, while this effect shrinks when the number of RIS elements is sufficiently large. Increasing the number of transmit antennas reduces the AN power if the eavesdropper is quite close to the transmitter, while improves it when the eavesdropper is far away.