Cross-Domain Recommendation (CDR) is an effective way to alleviate the cold-start problem. However, previous work severely ignores fairness and bias when learning the mapping function, which is used to obtain the representations for fresh users in the target domain. To study this problem, in this paper, we propose a Fairness-aware Cross-Domain Recommendation model, called FairCDR. Our method achieves user-oriented group fairness by learning the fairness-aware mapping function. Since the overlapping data are quite limited and distributionally biased, FairCDR leverages abundant non-overlapping users and interactions to help alleviate these problems. Considering that each individual has different influence on model fairness, we propose a new reweighing method based on Influence Function (IF) to reduce unfairness while maintaining recommendation accuracy. Extensive experiments are conducted to demonstrate the effectiveness of our model.