In Video-based Facial Expression Recognition (V-FER), models are typically trained on closed-set datasets with a fixed number of known classes. However, these V-FER models cannot deal with unknown classes that are prevalent in real-world scenarios. In this paper, we introduce a challenging Open-set Video-based Facial Expression Recognition (OV-FER) task, aiming at identifying not only known classes but also new, unknown human facial expressions not encountered during training. While existing approaches address open-set recognition by leveraging large-scale vision-language models like CLIP to identify unseen classes, we argue that these methods may not adequately capture the nuanced and subtle human expression patterns required by the OV-FER task. To address this limitation, we propose a novel Human Expression-Sensitive Prompting (HESP) mechanism to significantly enhance CLIP's ability to model video-based facial expression details effectively, thereby presenting a new CLIP-based OV-FER approach. Our proposed HESP comprises three components: 1) a textual prompting module with learnable prompt representations to complement the original CLIP textual prompts and enhance the textual representations of both known and unknown emotions, 2) a visual prompting module that encodes temporal emotional information from video frames using expression-sensitive attention, equipping CLIP with a new visual modeling ability to extract emotion-rich information, 3) a delicately designed open-set multi-task learning scheme that facilitates prompt learning and encourages interactions between the textual and visual prompting modules. Extensive experiments conducted on four OV-FER task settings demonstrate that HESP can significantly boost CLIP's performance (a relative improvement of 17.93% on AUROC and 106.18% on OSCR) and outperform other state-of-the-art open-set video understanding methods by a large margin.