Why is arginine's positive side chain classified as basic and not acidic?
Your textbook is imprecise. We consider whether a sidechain is acidic or basic based on the action of the sidechain when it is neutral.

Because arginine's sidechain is basic, it preferentially picks up a proton, and hence becomes positively charged. (In the same vein, carboxylic acid sidechains are acidic because they preferentially lose a proton, and exist as the negatively charged carboxylate anion.)