Temporal action segmentation (TAS) in untrimmed videos requires dense temporal supervision. However, most of the annotation cost is spent identifying action transitions where segmentation errors concentrate and small temporal shifts can disproportionately degrade segment-level metrics. We introduce B-ACT, a clip-budgeted active learning framework that explicitly allocates supervision to these error-prone boundary regions. B-ACT operates in a hierarchical two-stage loop: (i) it ranks and queries unlabeled videos using predictive uncertainty, and (ii) within each selected video, it detects candidate transitions from the current model predictions and selects the top-$K$ boundaries via a novel boundary score. The boundary score fuses neighborhood uncertainty, class ambiguity, and temporal prediction dynamics to reveal the underlying importance of each frame. Importantly, our annotation protocol requests labels only at the boundary frames while still training on boundary-centered clips to exploit temporal context through the model's receptive field. Extensive experiments on GTEA, 50Salads, and Breakfast demonstrate that boundary-centric supervision delivers strong label efficiency and consistently surpasses representative TAS active learning baselines and prior state of the art under sparse budgets. Gains are largest on datasets where performance is highly sensitive to boundary placement, as measured by edit and overlap-based F1 metrics.