Graph Neural Networks (GNNs) have shown exceptional success in graph representation learning and a wide range of real-world applications. However, scaling deeper GNNs poses challenges due to the neighbor explosion problem when training on large-scale graphs. To mitigate this, a promising class of GNN training algorithms utilizes historical embeddings to reduce computation and memory costs while preserving the expressiveness of the model. These methods leverage historical embeddings for out-of-batch nodes, effectively approximating full-batch training without losing any neighbor information-a limitation found in traditional sampling methods. However, the staleness of these historical embeddings often introduces significant bias, acting as a bottleneck that can adversely affect model performance. In this paper, we propose a novel VersatIle Staleness-Aware GNN, named VISAGNN, which dynamically and adaptively incorporates staleness criteria into the large-scale GNN training process. By embedding staleness into the message passing mechanism, loss function, and historical embeddings during training, our approach enables the model to adaptively mitigate the negative effects of stale embeddings, thereby reducing estimation errors and enhancing downstream accuracy. Comprehensive experiments demonstrate the effectiveness of our method in overcoming the staleness issue of existing historical embedding techniques, showcasing its superior performance and efficiency on large-scale benchmarks, along with significantly faster convergence.