joining a sorority with a boyfriend

No, they don't take their boyfriends to mixers; that would be catastrophic. Like you said, mixers are for socializing with frats and the guys in them. But those guys aren't there only for hooking up (I mean, a lot of them are). Socializing doesn't automatically mean hooking up. I'd say around half or more of the girls at mixers don't hook up with other guys at the mixer. Meet new guys, make friends; that's what mixers are about.

There are plenty of guys in frats with girlfriends, and they still go to mixers, and plenty of girls with boyfriends who go to mixers. They socialize and talk and have a good time, they just (generally) don't hook up.

People who consider mixers as a way to hook up will be disappointed; it's honestly way easier to hook up with someone at an open.

And you will absolutely not be looked down on for not hooking up with people at mixers. The dynamic is no different for a sister who hooks up vs. a sister who doesn't.