Recent advancements in large language models (LLMs) have demonstrated their remarkable capabilities across various language tasks. Inspired by the success of text-to-text translation refinement, this paper investigates how LLMs can improve the performance of speech translation by introducing a joint refinement process. Through the joint refinement of speech translation (ST) and automatic speech recognition (ASR) transcription via LLMs, the performance of the ST model is significantly improved in both training-free in-context learning and parameter-efficient fine-tuning scenarios. Additionally, we explore the effect of document-level context on refinement under the context-aware fine-tuning scenario. Experimental results on the MuST-C and CoVoST 2 datasets, which include seven translation tasks, demonstrate the effectiveness of the proposed approach using several popular LLMs including GPT-3.5-turbo, LLaMA3-8B, and Mistral-12B. Further analysis further suggests that jointly refining both transcription and translation yields better performance compared to refining translation alone. Meanwhile, incorporating document-level context significantly enhances refinement performance. We release our code and datasets on GitHub.