Making accurate recommendations for cold-start users has been a longstanding and critical challenge for recommender systems (RS). Cross-domain recommendations (CDR) offer a solution to tackle such a cold-start problem when there is no sufficient data for the users who have rarely used the system. An effective approach in CDR is to leverage the knowledge (e.g., user representations) learned from a related but different domain and transfer it to the target domain. Fine-tuning works as an effective transfer learning technique for this objective, which adapts the parameters of a pre-trained model from the source domain to the target domain. However, current methods are mainly based on the global fine-tuning strategy: the decision of which layers of the pre-trained model to freeze or fine-tune is taken for all users in the target domain. In this paper, we argue that users in RS are personalized and should have their own fine-tuning policies for better preference transfer learning. As such, we propose a novel User-specific Adaptive Fine-tuning method (UAF), selecting which layers of the pre-trained network to fine-tune, on a per-user basis. Specifically, we devise a policy network with three alternative strategies to automatically decide which layers to be fine-tuned and which layers to have their parameters frozen for each user. Extensive experiments show that the proposed UAF exhibits significantly better and more robust performance for user cold-start recommendation.