Close to zero is fine. Below zero is not. As long as you know exactly when and how much any charges to that account will be, you can plan for it and make sure the money is only there when it needs to be.
There was a time when I would manage my checking and savings accounts as you describe. However interest rates now don't make this type of micromanagement seem worthwhile to me. To make it work you really need to keep on top of bill due dates, pay days, etc. to make sure that you transfer as much as possible (but not too much) to the savings account at the appointed time, or make reverse transfers when needed. With a 1% difference between checking and savings interest rates, even keeping a $10,000 cushion in your checking account causes you to lose out on $100 per year (minus taxes). That's not nothing, but I find the freedom to just set all of my bills on auto-pay and not worry about the details to be worth at least that much.