You should be aware of the various resolutions and the sorts of devices people will use to view your site, and make that part of your test plan.
But given a responsive design, you want to check it on devices that you have reason to believe people are going to view it on, and use your understanding of your market/userbase to prioritise bugs that turn up. So in most cases, if you load up your site in a PC browser window and it breaks when you resize it, that's a bug and you'll have to fix it. If it doesn't look quite right on a few different popular smartphones and tablets going back a couple of years, those are probably a bugs and you'll probably want to fix them. If you get an opportunity to test on a Samsung A411 and it's all broken to hell, that is probably not worth fixing unless you have reason to believe a lot of people amongst your potential users will be using cheap legacy devices (but there was a time when that was 60% of my users, so I absolutely had to fix those bugs!).