Inductive Knowledge Graph Completion (KGC) aims to infer missing facts between newly emerged entities within knowledge graphs (KGs), posing a significant challenge. While recent studies have shown promising results in inferring such entities through knowledge subgraph reasoning, they suffer from (i) the semantic inconsistencies of similar relations, and (ii) noisy interactions inherent in KGs due to the presence of unconvincing knowledge for emerging entities. To address these challenges, we propose a Semantic Structure-aware Denoising Network (S$^2$DN) for inductive KGC. Our goal is to learn adaptable general semantics and reliable structures to distill consistent semantic knowledge while preserving reliable interactions within KGs. Specifically, we introduce a semantic smoothing module over the enclosing subgraphs to retain the universal semantic knowledge of relations. We incorporate a structure refining module to filter out unreliable interactions and offer additional knowledge, retaining robust structure surrounding target links. Extensive experiments conducted on three benchmark KGs demonstrate that S$^2$DN surpasses the performance of state-of-the-art models. These results demonstrate the effectiveness of S$^2$DN in preserving semantic consistency and enhancing the robustness of filtering out unreliable interactions in contaminated KGs.