Large language models (LLMs) are vulnerable to safety risks during fine-tuning, where small amounts of malicious or harmless data can compromise safeguards. In this paper, building on the concept of alignment direction -- defined by the weight difference between aligned and unaligned models -- we observe that perturbations along this direction preserve model safety. In contrast, perturbations along directions orthogonal to this alignment are strongly linked to harmful direction perturbations, rapidly degrading safety and framing the parameter space as a narrow safety basin. Based on this insight, we propose a methodology for safety fine-tuning called AsFT (Anchoring Safety in Fine-Tuning), which integrates a regularization term into the training objective. This term uses the alignment direction as an anchor to suppress updates in harmful directions, ensuring that fine-tuning is constrained within the narrow safety basin. Extensive experiments on multiple datasets show that AsFT outperforms Safe LoRA, reducing harmful behavior by 7.60 percent, improving model performance by 3.44 percent, and maintaining robust performance across various experimental settings. Code is available at https://github.com/PKU-YuanGroup/AsFT