We focus on learning adversarially robust classifiers under a cost-sensitive scenario, where the potential harm of different classwise adversarial transformations is encoded in a binary cost matrix. Existing methods are either empirical that cannot certify robustness or suffer from inherent scalability issues. In this work, we study whether randomized smoothing, a more scalable robustness certification framework, can be leveraged to certify cost-sensitive robustness. Built upon a notion of cost-sensitive certified radius, we show how to adapt the standard randomized smoothing certification pipeline to produce tight robustness guarantees for any cost matrix. In addition, with fine-grained certified radius optimization schemes specifically designed for different data subgroups, we propose an algorithm to train smoothed classifiers that are optimized for cost-sensitive robustness. Extensive experiments on image benchmarks and a real-world medical dataset demonstrate the superiority of our method in achieving significantly improved performance of certified cost-sensitive robustness while having a negligible impact on overall accuracy.