Simultaneously transmitting and reflecting reconfigurable intelligent surfaces (STAR-RISs) have been receiving great attention nowadays due to the capability of achieving full-space coverage. In this paper, the queue-aware STAR-RIS assisted non-orthogonal multiple access (NOMA) communication system is investigated to ensure the system stability. To tackle the challenge of infinite time periods required for stability, the long-term stability-oriented problem is reformulated as a queue-weighted sum rate (QWSR) maximization problem in each single time slot based on the Lyapunov drift theory. In particular, the rate weight allocated to each user is determined by the length of a data queue, which is maintained at the base station (BS) and pending delivery to each user. Then, the QWSR is maximized by jointly optimizing the NOMA decoding order, the active beamforming coefficients (ABCs) at the BS, and the passive transmission and reflection coefficients (PTRCs) at the STAR-RIS, where three STAR-RIS operating protocols are considered, namely energy splitting (ES), mode switching (MS), and time switching (TS). For ES, to handle the highly-coupled and non-convex problem, the blocked coordinate descent and the successive convex approximation methods are invoked to iteratively and alternatively optimize the problem. Moreover, the proposed iterative algorithm is further extended to a penalty-based two-loop algorithm to solve the binary amplitude constrained problem for MS. For TS, the formulated problem is decomposed into two subproblems, each of which can be solved in a similar manner as introduced for ES. Simulation results show that: i) our proposed STAR-RIS assisted NOMA communication achieves better performance compared with the conventional schemes; ii) the reformulated QWSR maximization problem is proven to ensure the system stability; and iii) the TS protocol achieves superior performance with respect to both the QWSR and the average queue length.