Yes. I don't think I've heard anyone raise the issue that it would hurt minorities more (though, it would prevent sales to convicted felons and minorities are overrepresented in that group).
The NRA opposes them. We do have background checks for most sales, but not for sales from individuals or at a gun show. The so-called gun show loophole gets a lot of attention, but I don't think it would do a damned thing to slow the mass shootings. I don't know that any of the shooters in the past several years have bought at a gun show (maybe the Vegas guy?) and I don't know how many of them would have been deterred by a background check.
Most Americans favor background checks. It's not at all clear to me how they're supposed to solve the rise in mass shootings.
I'm not saying that universal background checks wouldn't decrease deaths by gun, mind. I'm just saying that this topic comes up after every mass shooting as if it would reduce the number of these kinds of deaths and I don't see any obvious way it would have a substantial impact.
ETA: I see in the graphic in
this article that experts regard background checks for gun buyers would be one of the most effective remedies to mass shootings. It doesn't explain how it would be effective.