I doubt ArXiv considers the hosting of pirated content part of their mission or that they'd continue to host an article after receiving a valid DMCA takedown notice. In other words, I believe ArXiv depends on authors' restraining themselves from signing away their right to publish on ArXiv: physicists mostly engage in such restraint, but, e.g., chemists and medical researchers mostly do not.

ADDED. And over the course of ArXiv's existence, a significant fraction of authors have signed away the rights to the final post-peer-review version of their paper, which is why ArXiv has often been referred to as a preprint server.