Well, someone has to unload the basket - whether it is the cashier or the customer. So, being somewhat short myself (5'3") I gotta figure that if the counter is at a height which would make unloading difficult for the cashier, it's almost certainly going to be difficult for me too.
I very seldom use a basket (I almost always just use a cart on wheels) but when I do, and when I see other people do, I don't remember seeing them unload it. They (and I) just plop it down on the counter or conveyer belt. And in drug stores (IME), as others have noted, there isn't room on that small counter for both the basket and the unloaded items so am I supposed to continue holding the basket while unloading it onto the counter? I'm not gonna do that.
I think the problem here lies with the design of the cashier station. It's not done right and it's causing the cashier undue stress. And that issue is between the cashier and the employer. As the custumer, it's really not my concern if the store has designed the cashier station badly unless it affects me. And the only way it's going to affect me is if I'm required to unload my purchases from the basket and if there's potential for me to hurt myself in so doing.
I think we do tend to see things others do as rude when it affects us in a negative manner as in this case, even though it's really not rude at all. It seems to me that just based on how many people responded to this thread with opposing habits of unloading vs not, there really is no accepted SOP for these hand-held baskets. So I don't think these people are rude. I think it's a case of - some people do it one way, some do it the other.
The real issue is the set up and that is something the OP has to work with the store management/ownder to resolve.