In real-world classification settings, individuals respond to classifier predictions by updating their features to increase their likelihood of receiving a particular (positive) decision (at a certain cost). Yet, when different demographic groups have different feature distributions or different cost functions, prior work has shown that individuals from minority groups often pay a higher cost to update their features. Fair classification aims to address such classifier performance disparities by constraining the classifiers to satisfy statistical fairness properties. However, we show that standard fairness constraints do not guarantee that the constrained classifier reduces the disparity in strategic manipulation cost. To address such biases in strategic settings and provide equal opportunities for strategic manipulation, we propose a constrained optimization framework that constructs classifiers that lower the strategic manipulation cost for the minority groups. We develop our framework by studying theoretical connections between group-specific strategic cost disparity and standard selection rate fairness metrics (e.g., statistical rate and true positive rate). Empirically, we show the efficacy of this approach over multiple real-world datasets.