Foundation Models (FMs) are gaining increasing attention in the biomedical artificial intelligence (AI) ecosystem due to their ability to represent and contextualize multimodal biomedical data. These capabilities make FMs a valuable tool for a variety of tasks, including biomedical reasoning, hypothesis generation, and interpreting complex imaging data. In this review paper, we address the unique challenges associated with establishing an ethical and trustworthy biomedical AI ecosystem, with a particular focus on the development of FMs and their downstream applications. We explore strategies that can be implemented throughout the biomedical AI pipeline to effectively tackle these challenges, ensuring that these FMs are translated responsibly into clinical and translational settings. Additionally, we emphasize the importance of key stewardship and co-design principles that not only ensure robust regulation but also guarantee that the interests of all stakeholders-especially those involved in or affected by these clinical and translational applications-are adequately represented. We aim to empower the biomedical AI community to harness these models responsibly and effectively. As we navigate this exciting frontier, our collective commitment to ethical stewardship, co-design, and responsible translation will be instrumental in ensuring that the evolution of FMs truly enhances patient care and medical decision-making, ultimately leading to a more equitable and trustworthy biomedical AI ecosystem.