Sepsis is a syndrome characterized by a dysregulated immune response to infection, which often leads to acute respiratory distress syndrome (ARDS) and the need for mechanical ventilation (MV) among patients in intensive care units (ICUs), which substantially affect patient morbidity and mortality. Although existing studies have attempted to predict the risk of ARDS or MV individually, joint modeling of both as multiple outcomes remains relatively rare. This is partly because prior research has not sufficiently explored the potential interaction between ARDS and MV, thereby overlooking their close relationship in the pathological progression of sepsis. Based on this, we propose a joint prediction model constructed using a multi-task neural network based on long short-term memory (LSTM), specifically designed to simultaneously predict the risk of ARDS onset and the need for MV in ICU patients with sepsis. The model integrates an LSTM-based feature extraction module, a feature fusion component, and task-specific prediction heads, allowing it to effectively capture and integrate task-relevant temporal patterns. We validated the model on the MIMIC-IV database. The results demonstrate that it achieved an AUROC of 0.82 for ARDS prediction and 0.91 for MV prediction. Additional experiments further verified the model's stability and effectiveness. Moreover, we employed SHapley Additive exPlanations (SHAP) to interpret the model's outputs, offering insights into clinically relevant features and supporting personalized treatment planning. In summary, the model enables efficient and interpretable joint prediction of ARDS and MV, providing a useful decision-support tool for the early identification of high-risk sepsis patients in critical care environments.