But no, seriously, it depends on what kind of set theory you're talking about. The stuff Hilbert was arguing about up to the stuff still being argued about today in the ivory tower? No, you pragmatically do not need to know that unless you're pursuing a research career in pure math. The stuff in an undergrad probability course? Yeah, sets come up basically everywhere if you're looking and are willing to think like that.
Modern pure mathematicians benefit from being able to conduct most of their reasoning on the lower footsteps of the von Neumann hierarchy (V_{omega + k} for small k), where they get access to higher-order tools like function spaces and topologies, countability as a fruitful property to impose and work with, and a playground where most of the disparate branches of mathematical research can be brought together without worrying about bindings between underlying frameworks. None of these things require a deep knowledge of the metamath of ZF to work with, but they'd all make many 19th-century mathematicians nervous.
Note also that the most frequently used tools of naive set theory today (e.g. unions/intersections, separations/replacements, powersets, cartesian products/relations/functions, a few notable infinite sets) roughy mirror ZF. That shouldn't seem like an accident. Having working mathematicians trained to think and organize their work in these constructs keeps them around those lower footsteps.