Is there a basis for the arm and leg wraps? In Naruto, many characters have what appear to be bandages wrapped around their arms or legs. Is there a basis for this in any martial arts or military fighting styles?
These originate from a similar practice in the real world, where they are used in boxing, combat sports, and martial arts (largely muay thai, one of the more brutal sports). They are mostly seen on the wrists, but can be used on the legs too for the same reasons. (Legs are sometimes wrapped around the hips and thighs, but can be wrapped lower down as well.)

There are different types of wraps: cloth/elastic, gauze, or a type of cast material which is much firmer.

Cloth wraps help protect from superficial cuts and other scrapes; you may have seen (or been in) situations where, after a punch, your knuckles can get worn down or bloodied up as the skin dries/breaks from the impact. Having a cover over them would obviously protect them from doing so.

Gauze and stronger cast materials (and even cloth, to a lesser extent) are for protecting the wrist and hand from damage sustained while punching. They help by holding the hand in its proper shape by wrapping around the fingers and the thumb and keeping the structure intact when an impact is sustained.


(Source: Wikipedia)

It is more than likely that Naruto adopted this to have their characters appear as (semi-)realistic, prepared martial artists.