Document-level context is crucial for handling discourse challenges in text-to-text document-level machine translation (MT). Despite the increased discourse challenges introduced by noise from automatic speech recognition (ASR), the integration of document-level context in speech translation (ST) remains insufficiently explored. In this paper, we develop DoCIA, an online framework that enhances ST performance by incorporating document-level context. DoCIA decomposes the ST pipeline into four stages. Document-level context is integrated into the ASR refinement, MT, and MT refinement stages through auxiliary LLM (large language model)-based modules. Furthermore, DoCIA leverages document-level information in a multi-level manner while minimizing computational overhead. Additionally, a simple yet effective determination mechanism is introduced to prevent hallucinations from excessive refinement, ensuring the reliability of the final results. Experimental results show that DoCIA significantly outperforms traditional ST baselines in both sentence and discourse metrics across four LLMs, demonstrating its effectiveness in improving ST performance.