Recently, simultaneously transmitting and reflecting reconfigurable intelligent surfaces (STAR-RISs) have received significant research interest. The employment of large STAR-RIS and high-frequency signaling inevitably make the near-field propagation dominant in wireless communications. In this work, a STAR-RIS aided near-field multiple-input multiple-multiple (MIMO) communication framework is proposed. A weighted sum rate maximization problem for the joint optimization of the active beamforming at the base station (BS) and the transmission/reflection-coefficients (TRCs) at the STAR-RIS is formulated. The non-convex problem is solved by a block coordinate descent (BCD)-based algorithm. In particular, under given STAR-RIS TRCs, the optimal active beamforming matrices are obtained by solving a convex quadratically constrained quadratic program. For given active beamforming matrices, two algorithms are suggested for optimizing the STAR-RIS TRCs: a penalty-based iterative (PEN) algorithm and an element-wise iterative (ELE) algorithm. The latter algorithm is conceived for STAR-RISs with a large number of elements. Numerical results illustrate that: i) near-field beamforming for STAR-RIS aided MIMO communications significantly improves the achieved weighted sum rate compared with far-field beamforming; ii) the near-field channels facilitated by the STAR-RIS provide enhanced degrees-of-freedom and accessibility for the multi-user MIMO system; and iii) the BCD-PEN algorithm achieves better performance than the BCD-ELE algorithm, while the latter has a significantly lower computational complexity.