While current AI systems represent significant technological achievements, their success in large-scale, low-cost automation of audio processing tasks comes with a number of documented and potential harms. As a result, researchers have proposed a variety of algorithmic interventions -- technical methods that disrupt, identify, constrain, or otherwise influence the operation of AI systems directly to mitigate harms -- with mixed results. This dissertation asks three questions: where and why have existing algorithmic interventions fallen short? Where have they shown promise? And how can they be made more effective going forward? To answer, I present original research on three classes of algorithmic intervention: adversarial interventions that interfere with irresponsibly deployed AI systems to protect the privacy and intellectual property of those interacting with them; cooperative interventions that operate alongside deployed AI systems with the support of developers or digital media platforms to identify or constrain AI outputs; and design interventions that incorporate governance and harm-mitigation objectives directly into the development of AI systems. Through this work, I argue that algorithmic interventions can be made more effective by moving from reactive, extrinsic measures that sit outside of audio AI systems ("token fixes") to proactive, intrinsic measures that shape the design and behavior of audio AI systems.