Natural language processing (NLP) models often replicate or amplify social bias from training data, raising concerns about fairness. At the same time, their black-box nature makes it difficult for users to recognize biased predictions and for developers to effectively mitigate them. While some studies suggest that input-based explanations can help detect and mitigate bias, others question their reliability in ensuring fairness. Existing research on explainability in fair NLP has been predominantly qualitative, with limited large-scale quantitative analysis. In this work, we conduct the first systematic study of the relationship between explainability and fairness in hate speech detection, focusing on both encoder- and decoder-only models. We examine three key dimensions: (1) identifying biased predictions, (2) selecting fair models, and (3) mitigating bias during model training. Our findings show that input-based explanations can effectively detect biased predictions and serve as useful supervision for reducing bias during training, but they are unreliable for selecting fair models among candidates.Our code is available at https://github.com/Ewanwong/fairness_x_explainability.