Biomedical Event Extraction (BEE) is a challenging task that involves modeling complex relationships between fine-grained entities in biomedical text. BEE has traditionally been formulated as a classification problem. With the recent technological advancements in large language models (LLMs), generation-based models that cast event extraction as a sequence generation problem have attracted much attention from the NLP research communities. However, current generative models often overlook the importance of cross-instance information from complex event structures such as nested events and overlapping events, which contribute to over 20% of the events in the benchmark datasets. In this paper, we propose an event structure-aware generative model named GenBEE, which can capture complex event structures in biomedical text for biomedical event extraction. In particular, GenBEE constructs event prompts that distill knowledge from LLMs for incorporating both label semantics and argument dependency relationships into the proposed model. In addition, GenBEE also generates prefixes with event structural prompts to incorporate structural features for improving the model's overall performance. We have evaluated the proposed GenBEE model on three widely used biomedical event extraction benchmark datasets, namely MLEE, GE11, and PHEE. Experimental results show that GenBEE has achieved state-of-the-art performance on the MLEE and GE11 datasets, and achieved competitive results when compared to the state-of-the-art classification-based models on the PHEE dataset.