My mother lives in an in law apt attached to our home and it's worked out well for all of us. It hasn't been perfect, but it was easier that way than it was when she was living an hour away. She's been here 25 yrs. I think she's always benefitted from being here, but especially now, as she is ill, and has two personal caregivers here (myself and DD are nurses) and all of her needs taken care of. It is a family affair.
The one thing I will tell you is that when we built it, we had an architect design it as handicap accessible. For many years, that didn't matter - until the past year or so. Now, we're so thankful we did that. We had an OT and PT come in when Mom was in rehab and they said it's rare to see the set up we had for safety, and that lots of people have regrets when they can't go home because they're not safe there. So I'd say for a basement apartment that's something you need to really think about. 1) stairs, 2) dampness, 3) fire and CO, etc. Your city or town may have ordinances about having a stove, too, which you should check out if you have to do it illegally, for insurance coverage.
I think a lot of us are picturing a dank basement, but some basements can be really lovely. Ours is only half in the ground, with windows and natural light. It can still get a little damp, though. Every set up is different. Good luck with your decision. I live in a high COL area and this has become pretty common here. My in laws did it, too, and lots of friends and other family.