An active-sensing-based sense-then-train (STT) scheme is proposed for beam training in near-field multiple-input multiple-output (MIMO) systems. Compared to conventional codebook-based schemes, the proposed STT scheme is capable of not only addressing the complex spherical-wave propagation but also effectively exploiting the additional degrees-of-freedoms (DoFs). The STT scheme is tailored for both single-beam and multi-beam cases. 1) For the single-beam case, the STT scheme first utilizes a sensing phase to estimate a low-dimensional representation of the near-field MIMO channel in the truncated wavenumber domain. Then, in the subsequent training phase, the neural network modules at transceivers are updated online to align beams, utilizing sequentially received ping-pong pilots. This approach can efficiently obtain the aligned beam pair without relying on predefined codebooks or training datasets. 2) For the multi-beam case, based on the single-beam STT, a Gram-Schmidt method is further utilized to guarantee the orthogonality between beams in the training phase. Numerical results unveil that 1) the proposed STT scheme can significantly enhance the beam training performance in the near field compared to the conventional far-field codebook-based schemes, and 2) the proposed STT scheme can perform fast and low-complexity beam training, while achieving a near-optimal performance without full channel state information in both cases.