I don't give partial credit for work only partially completed—mostly I won't take it unless it's complete. I do offer partial credit (a "C" with correct answers) for work turned in a day late, so kids can still keep the grade up. As an example, I tell my kids that when they start working, if they don't stay the whole day, they may get paid for only a couple of hours, but repeatedly doing so will get them fired.
I don't give busy work for homework. I try to make it meaningful, manageable, and even fun. Some kids love it. Some make do. And some refuse to do anything—for those kids, yes, their grades are bad. If they did the homework when they can use notes and a book, without the pressure of "testing," they can keep lower test grades from destroying their class average, but actually it should help keep their test grades up: often it's review or practice.
I won't stop giving homework because kids refuse to do it. If it's reasonable work for the age/level and relevant, then they should do it. It's like the boss asking you to write a report: you just to it.
And as a parent and teacher, I expect my daughter to do her best. It was the same with the older two before they graduated. If they needed help, we were there. If the youngest needs help, it's never a problem. But my daughter is not only learning content and reinforcing what she learns (and lots of times she hates it), she is learning to be responsible and stick with it even if she doesn't like it. If she doesn't do it, those grades will bite her in the butt, and it will be a situation of her making.
I believe the same goes for the classroom. If someone has a problem at home, or is struggling with a concept, I am there. I don't need to know their business, but if they explain there is a crisis at home, I can work something out with them. Otherwise, if they pay attention in class and follow directions, the homework isn't impossible. I don't want to beat them into the ground; I just want them to leave me knowing more than when they arrived.