Pre-trained large language models (LLMs) have been reliably integrated with visual input for multimodal tasks. The widespread adoption of instruction-tuned image-to-text vision-language assistants (VLAs) like LLaVA and InternVL necessitates evaluating gender biases. We study gender bias in 22 popular open-source VLAs with respect to personality traits, skills, and occupations. Our results show that VLAs replicate human biases likely present in the data, such as real-world occupational imbalances. Similarly, they tend to attribute more skills and positive personality traits to women than to men, and we see a consistent tendency to associate negative personality traits with men. To eliminate the gender bias in these models, we find that fine-tuning-based debiasing methods achieve the best trade-off between debiasing and retaining performance on downstream tasks. We argue for pre-deploying gender bias assessment in VLAs and motivate further development of debiasing strategies to ensure equitable societal outcomes. Code is available at https://github.com/ExplainableML/vla-gender-bias.