Cross-Domain Recommendation (CDR) is an effective approach for mitigating the cold-start problem. However, previous research largely overlooks the issue of fairness: undesired bias associated with sensitive attributes in CDR scenarios. Specifically, through our data-driven experiments, we find that existing CDR-based methods are particularly vulnerable to the distribution bias in sparse cross-domain data, and such bias easily results in unfair treatment of cold-start users in the target domain, eventually exacerbating the recommendation feedback loop. Additionally, existing fairness-aware recommendation algorithms mainly focus on scenarios where unfairness arises during the “warm-start” phase. However, there is a lack of research dedicated to addressing user unfairness during the “cold-start” phase. To tackle the above problems, in this paper, we propose a novel fairness-aware algorithm for CDR, called FairCDR. Specifically, FairCDR granularly models the individual influence of each sample on both fairness and accuracy to facilitate data reweighting, effectively striking a balance between fairness and recommendation utility.Extensive experiments are conducted to demonstrate the effectiveness of our proposed method. To benefit the research community, we have released our project at https://faircdr.github.io/FairCDR/