Is sex work (specifically, prostitution) vulnerable to technological unemployment? Several authors have argued that it is. They claim that the advent of sophisticated sexual robots will lead to the displacement of human prostitutes, just as, say, the advent of sophisticated manufacturing robots have displaced many traditional forms of factory labour. But are they right? In this article, I critically assess the argument that has been made in favour of this displacement hypothesis. Although I grant the argument a degree of credibility, I argue that the opposing hypothesis -- that prostitution will be resilient to technological unemployment -- is also worth considering. Indeed, I argue that increasing levels of technological unemployment in other fields may well drive more people into the sex work industry. Furthermore, I argue that no matter which hypothesis you prefer -- displacement or resilience -- you can make a good argument for the necessity of a basic income guarantee, either as an obvious way to correct for the precarity of sex work, or as a way to disincentivise those who may be drawn to prostitution.