Recent advances in deep learning research have shown remarkable achievements across many tasks in computer vision (CV) and natural language processing (NLP). At the intersection of CV and NLP is the problem of image captioning, where the related models' robustness against adversarial attacks has not been well studied. This paper presents a novel adversarial attack strategy, AICAttack (Attention-based Image Captioning Attack), designed to attack image captioning models through subtle perturbations on images. Operating within a black-box attack scenario, our algorithm requires no access to the target model's architecture, parameters, or gradient information. We introduce an attention-based candidate selection mechanism that identifies the optimal pixels to attack, followed by a customised differential evolution method to optimise the perturbations of pixels' RGB values. We demonstrate AICAttack's effectiveness through extensive experiments on benchmark datasets against multiple victim models. The experimental results demonstrate that our method outperforms current leading-edge techniques by achieving consistently higher attack success rates.