Most imitation learning planners for autonomous driving are trained with displacement-based objectives such as average displacement error (ADE), which favor average accuracy but overlook how often predicted trajectories become unsafe. CARE Planner augments an attention-constrained Transformer with a Conditional Value at Risk (CVaR) based risk module that measures clearance-based tail risk along the prediction horizon. This risk is used both to select the supervised trajectory mode and to construct a tail-risk-aware soft target that downweights unsafe modes during multimodal learning. Compared to state-of-the-art planners such as PlanTF and our previous framework CAR Planner, CARE Planner outperforms overall performance and significant improvements in safety-related metrics on the nuPlan benchmark, highlighting the effectiveness of risk-aware training in enhancing the reliability of imitation learning planners.