Graph Neural Networks (GNNs) are powerful in learning semantics of graph data. Recently, a new paradigm "pre-train, prompt" has shown promising results in adapting GNNs to various tasks with less supervised data. The success of such paradigm can be attributed to the more consistent objectives of pre-training and task-oriented prompt tuning, where the pre-trained knowledge can be effectively transferred to downstream tasks. However, an overlooked issue of existing studies is that the structure information of graph is usually exploited during pre-training for learning node representations, while neglected in the prompt tuning stage for learning task-specific parameters. To bridge this gap, we propose a novel structure-based prompting method for GNNs, namely SAP, which consistently exploits structure information in both pre-training and prompt tuning stages. In particular, SAP 1) employs a dual-view contrastive learning to align the latent semantic spaces of node attributes and graph structure, and 2) incorporates structure information in prompted graph to elicit more pre-trained knowledge in prompt tuning. We conduct extensive experiments on node classification and graph classification tasks to show the effectiveness of SAP. Moreover, we show that SAP can lead to better performance in more challenging few-shot scenarios on both homophilous and heterophilous graphs.