Yeah, I would. Personally, I get along with my sisters and could easily say honest and kind things about them. But even if I didn’t like them much, I’d still want them to have work. You know your sister best. Maybe she’s wealthy and just looking for a side gig – in which case saying nothing would be fine. But if she’s struggling, like so many people right now, missing out on the chance to help a family member would not align with my values. Depending on where you are, being employed could be the difference between getting a fatal illness treated or not. It’s unsafe not to be employed, as opposed to merely inconvenient.
Being a family member does not necessarily mean you can’t provide a reference. And shoving you, her sister, while awful for you, does not mean that she’s a threat to the public. It doesn’t sound like she has diagnosed mental health issues, but she’d need a job in that case, too! Many minor mental health issues can make people short-tempered. Mild depression, grief, burnout, anxiety, etc. If we excluded everyone with mental health problems from the job market, society would collapse.
You say that you don’t know anything about the job. For all you know, she’d be transcribing old documents from home, only interacting with her employer once a week for a quick phone call. Maybe she’d work as a gardener in the local cemetery, and that peaceful work ends up helping her heal from the mental health issues. There are lots of jobs like that. And if you still have concerns after finding out what the job is (being around guns, drugs, or violence should probably be reserved for our most stable people), you can just say goodbye and hang up.
By the way: I wouldn’t be so sure that she knowingly put you down as a reference. I’ve had to provide siblings’ names on job application forms before. I thought it was extremely weird, but some hiring processes are very strange. If she did mean to make you her reference, though, that’s a compliment! It means she trusts you.
Providing a reference is really not such a big deal. It’s not like being vetted to serve on the Supreme Court. In all likelihood, you’ll be talking to the guy for just a few minutes. She almost certainly had to put down other people as professional references, and you might be the “personal” one, just somebody from her social circle who can verify that she’s reliable and hard-working, or whatever she’s claimed. If there’s anything nice you can say about her, that could be very valuable in this situation.