AI-driven chatbots are seen as an attractive solution to support people undergoing emotional distress. One of the main components of such a chatbot is the ability to empathize with the user. But a significant limitation in achieving this goal is the lack of a large dialogue dataset containing empathetic support for those undergoing distress. In this work, we curate a large-scale dialogue dataset that contains ≈1.3M peer support dialogues spanning across more than 4K distress-related topics. We analyze the empathetic characteristics of this dataset using statistical and visual means. To demonstrate the utility of this dataset, we train four baseline neural dialogue models that can respond empathetically to distress prompts. Two of the baselines adapt existing architecture and the other two incorporate a framework identifying levels of cognitive and emotional empathy in responses. Automatic and human evaluation of these models validate the utility of the dataset in generating empathetic responses for distress support and show that identifying levels of empathy in peer-support responses facilitates generating responses that are lengthier, richer in empathy, and closer to the ground truth.