Cosmic strings that are attached to rapidly spinning black holes can extract significant amounts of rotational energy and angular momentum. Here we study the effect on primordial black holes, which are expected to form with one or more cosmic strings attached. Although large primordial black holes are predicted to rapidly spin up due to accretion soon after forming, we argue that cosmic strings will spin them down again. We show that if there are cosmic strings with tension greater than ${10}^{-20}$, the spins of large primordial black holes of mass greater than $30M_{\odot }$ should consequently be observed to be near zero. We also investigate the effect on a supermassive black hole of capturing a cosmic string and the possibility of observing the subsequent spin down by its effect on a pulsar orbiting the black hole.