Semi-supervised segmentation methods have demonstrated promising results in natural scenarios, providing a solution to reduce dependency on manual annotation. However, these methods face significant challenges when directly applied to pathological images due to the subtle color differences between nuclei and tissues, as well as the significant morphological variations among nuclei. Consequently, the generated pseudo-labels often contain much noise, especially at the nuclei boundaries. To address the above problem, this paper proposes a boundary-aware contrastive learning network to denoise the boundary noise in a semi-supervised nuclei segmentation task. The model has two key designs: a low-resolution denoising (LRD) module and a cross-RoI contrastive learning (CRC) module. The LRD improves the smoothness of the nuclei boundary by pseudo-labels denoising, and the CRC enhances the discrimination between foreground and background by boundary feature contrastive learning. We conduct extensive experiments to demonstrate the superiority of our proposed method over existing semi-supervised instance segmentation methods.