High-definition (HD) map construction methods are crucial for providing precise and comprehensive static environmental information, which is essential for autonomous driving systems. While Camera-LiDAR fusion techniques have shown promising results by integrating data from both modalities, existing approaches primarily focus on improving model accuracy and often neglect the robustness of perception models, which is a critical aspect for real-world applications. In this paper, we explore strategies to enhance the robustness of multi-modal fusion methods for HD map construction while maintaining high accuracy. We propose three key components: data augmentation, a novel multi-modal fusion module, and a modality dropout training strategy. These components are evaluated on a challenging dataset containing 10 days of NuScenes data. Our experimental results demonstrate that our proposed methods significantly enhance the robustness of baseline methods. Furthermore, our approach achieves state-of-the-art performance on the clean validation set of the NuScenes dataset. Our findings provide valuable insights for developing more robust and reliable HD map construction models, advancing their applicability in real-world autonomous driving scenarios. Project website: https://robomap-123.github.io.