“On a case-by-case basis.” “On a regular basis.” “On an urgent basis.”
Each of those base expressions, from The Associated Press Stylebook, no less, can be said differently, more fluidly: “Case by case.” “Regularly.” “Daily.”
There’s nothing grammatically wrong with those “on a (whatever) basis” phrases, except that they’re wordy. Or, as Bryan A. Garner puts it in his Modern American Usage, “The word basis often signals verbosity in adverbial constructions.” (Let’s hope his verbosity was ironic.)
In one week’s worth of Nexis citations from US publications and blogs, “on a case-by-case basis” appeared more than 100 times. “On a daily basis” appeared a whopping 600 times, give or take a few. If they had all been whittled down, more than 3,000 words could have been saved, or used to write several more articles.
Other uses of “basis” can be off base, or on target: “On a legal basis” could be just “legally,” but it would be fine to say, “That is the legal basis for my argument.” A “basis point” in financial contexts is a lot shorter than saying “one-thousandth of a point.” But watch out for “on a part-time basis,” “on a yearly basis,” and other such “on a basis”es, er, bases.
Now you’re on a first-name basis with basis.