Graph machine learning (GML) has made great progress in node classification, link prediction, graph classification and so on. However, graphs in reality are often structurally imbalanced, that is, only a few hub nodes have a denser local structure and higher influence. The imbalance may compromise the robustness of existing GML models, especially in learning tail nodes. This paper proposes a selective graph augmentation method (SAug) to solve this problem. Firstly, a Pagerank-based sampling strategy is designed to identify hub nodes and tail nodes in the graph. Secondly, a selective augmentation strategy is proposed, which drops the noisy neighbors of hub nodes on one side, and discovers the latent neighbors and generates pseudo neighbors for tail nodes on the other side. It can also alleviate the structural imbalance between two types of nodes. Finally, a GNN model will be retrained on the augmented graph. Extensive experiments demonstrate that SAug can significantly improve the backbone GNNs and achieve superior performance to its competitors of graph augmentation methods and hub/tail aware methods.