Drug-drug interaction (DDI) prediction is pivotal for drug safety and clinical decision-making. Recently, subgraph-based methods utilizing knowledge graphs (KGs) and domain information have achieved promising results by extracting informative subgraphs for DDI prediction. However, existing subgraph extraction methods are typically coarse-grained and nonspecific, facing two key limitations: First, they are constrained by the vast and noisy nature of real-world KGs, making it challenging to identify the most informative substructures from the massive space of candidate subgraphs. Second, current methods often fail to exploit the molecular structural specificity of drugs to selectively extract relevant subgraphs, lacking effective integration of molecular structure information with knowledge graph context. To address these challenges, we propose RISE-DDI, a novel framework for Reinforced-based Informative Subgraph Extraction approach for drug-drug interaction prediction. Specifically, RISE-DDI formulates the subgraph extraction as a Markov Decision Process (MDP) and leverages a deep reinforcement learning (RL) agent to dynamically and adaptively extract the most informative and context-specific subgraphs for each drug pair. The agent is guided by a learnable structure-aware reward model that considers both the topological context from the knowledge graph and the molecular features of the drug pairs, thereby encouraging the selection of subgraphs that are both structurally relevant and biologically informative. Extensive experiments on DDI benchmark datasets demonstrate that our method outperforms state-of-the-art baselines in both transductive and inductive scenarios, achieving improvements of up to 20\%. Furthermore, visualization analyses of the extracted subgraphs highlight the interpretability of our model, providing insights into the underlying mechanisms of drug interactions.