This paper investigates the downlink power allocation of the simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS)-assisted cell-free massive multiple-input multiple-output (MIMO) system with multi-antenna users. We introduce downlink spectral efficiency (SE) and derive novel closed-form SE expressions using linear minimum mean squared error (MMSE) detectors. We also address the downlink power allocation via a sum SE maximization problem framed within an alternating direction method of multipliers (ADMM)-based fractional programming (FP) algorithm. Numerical results demonstrate that systems utilizing multi-antenna users significantly enhance SE, achieving at least a 20% SE increase as the number of antennas increases from one to six. Additionally, our proposed ADMM-based FP algorithm outperforms existing fractional power control approaches, yielding a more than 20% SE increase. These results highlight the necessity for adopting multi-antenna users and efficient power allocation algorithms in STAR-RIS-assisted cell-free massive MIMO systems.