The proliferation of large AI models trained on uncurated, often sensitive web-scraped data has raised significant privacy concerns. One of the concerns is that adversaries can extract information about the training data using privacy attacks. Unfortunately, the task of removing specific information from the models without sacrificing performance is not straightforward and has proven to be challenging. We propose a rather easy yet effective defense based on backdoor attacks to remove private information, such as names and faces of individuals, from vision-language models by fine-tuning them for only a few minutes instead of re-training them from scratch. Specifically, by strategically inserting backdoors into text encoders, we align the embeddings of sensitive phrases with those of neutral terms-"a person" instead of the person's actual name. For image encoders, we map individuals' embeddings to be removed from the model to a universal, anonymous embedding. The results of our extensive experimental evaluation demonstrate the effectiveness of our backdoor-based defense on CLIP by assessing its performance using a specialized privacy attack for zero-shot classifiers. Our approach provides a new "dual-use" perspective on backdoor attacks and presents a promising avenue to enhance the privacy of individuals within models trained on uncurated web-scraped data.