There’s plenty of cheap PSX-to-USB adapters available for cheap. Can get some of the more cheaper ones off ebay for like $2-$5 and they (should) work just fine. Only rare exceptions I know of is when you’re using non-standard controllers (Taiko drums, dance pads, etc), and in that case, a more expensive adapter converter may be needed.
PSX-to-PS3 adapters should also work for a PC (they still use USB; I have one that works fine). I can’t imagine it being incompatible, but if ordering such an adapter and in-doubt, you should probably just order a cheap one.
Regardless of what adapter you get though, it should just be plug-and-play compatible. The only time you may need a driver is either for force-feedback (you usually don’t though), or non-standard controllers (for dance pads, you may need to enter a dance pad code to fix an axis issue which would only work with that adapter’s driver).