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Nucleus segmentation is an important task in medical image analysis. However, machine learning models cannot perform well because there are large amount of clusters of crowded nuclei. To handle this problem, existing approaches typically resort to sophisticated hand-crafted post-processing strategies; therefore, they are vulnerable to the variation of post-processing hyper-parameters. Accordingly, in this paper, we devise a Boundary-assisted Region Proposal Network (BRP-Net) that achieves robust instance-level nucleus segmentation. First, we propose a novel Task-aware Feature Encoding (TAFE) network that efficiently extracts respective high-quality features for semantic segmentation and instance boundary detection tasks. This is achieved by carefully considering the correlation and differences between the two tasks. Second, coarse nucleus proposals are generated based on the predictions of the above two tasks. Third, these proposals are fed into instance segmentation networks for more accurate prediction. Experimental results demonstrate that the performance of BRP-Net is robust to the variation of post-processing hyper-parameters. Furthermore, BRP-Net achieves state-of-the-art performances on both the Kumar and CPM17 datasets. The code of BRP-Net will be released at https://github.com/csccsccsccsc/brpnet.