Emotional support dialogue systems aim to reduce help-seekers' distress and help them overcome challenges. While human values$\unicode{x2013}$core beliefs that shape an individual's priorities$\unicode{x2013}$are increasingly emphasized in contemporary psychological therapy for their role in fostering internal transformation and long-term emotional well-being, their integration into emotional support systems remains underexplored. To bridge this gap, we present a value-driven method for training emotional support dialogue systems designed to reinforce positive values in seekers. Notably, our model identifies which values to reinforce at each turn and how to do so, by leveraging online support conversations from Reddit. We evaluate the method across support skills, seekers' emotional intensity, and value reinforcement. Our method consistently outperforms various baselines, effectively exploring and eliciting values from seekers. Additionally, leveraging crowd knowledge from Reddit significantly enhances its effectiveness. Therapists highlighted its ability to validate seekers' challenges and emphasize positive aspects of their situations$\unicode{x2013}$both crucial elements of value reinforcement. Our work, being the first to integrate value reinforcement into emotional support systems, demonstrates its promise and establishes a foundation for future research.