It may not be the only criterion of use, but "do we have good reason to believe they will succeed in college?" is at the very least a critical bar that ought to be passed. It's can be the difference between college as the empowering, education experience it's supposed to be, and college as the meat-grinder wealth-extraction tool it can be when you feed in students who shouldn't be there.
[1]: Not necessarily "best", but being "earliest" in a world of compounding interest makes them quite valuable in their own way.
In practice, that's not who we select. One college in the Midwest I know of selected 1) kids related to alumni, 2) people of a certain religion, 3) people who would add to the sports and music programs, 4) kids of color with good grades and high SAT scores, and last 5) people who didn't fit the above, especially from California, Texas, or Florida, who could pay full tuition. A major goal at some small colleges is balancing kids who will pay full price and kids who will do well at college, which creates a funny situation where you've got a well off heavy-majority-white party crowd with low test scores and a smaller and more diverse group of subsidized kids who bring many academic and non-academic talents (the music/sports/math/science/high school leadership crowd that may not have well-off parents).
People apply, pay for a year or two, inevitably fail, but are now also 30k in debt.
What do you think those people will do?
Accepting anyone only works in countries without paid universities.
Match students to the colleges that will be hard enough to get them to grow, but not so hard that they fail.
Well, the underlying basis is the finite number of seats for students. If you have a limited number of slots, you must reject some applicants. Therefore, the question flipped around is, "what criteria to use to reject applicants?"
That leads into what I guess your usage of "basis" is actually about: the philosophy of selecting on the "doing well" instead of some other criteria such as... "economic disadvantages?"... or "local student over out-of-state or foreign student?"..."athletic abilities?", etc. You'd have to expand on what alternatives you had in mind.
I think the "doing well" criteria can be defended by filling up the limited seats with students who will get the most out of college. For example, a student taking "Physics I" needs to know what derivatives are (Calculus). Having some slots used up by students that should have been in remedial math courses (Algebra) instead of flunking out of PHYICS1 is unfair to the potential students who already have the baseline math skills required for solving physics equations. Multiply that example across the entire curriculum required for a degree and the "doing well" criteria seems justified. You want to reject potential students who are not able to succeed at college-level course work. Maybe tests like SAT are bad/flawed but so far, they appear to be the "least bad" tool to predict which students will do well in college.
One can create institutions that don't care about the assessment of "doing well" as an entrance criteria as long as the students pay the fees. That signal is provided by many colleges[1] that don't require SAT/ACT scores. However, the prestige and respect of those institutions don't match the ones that do. That negative perception in the marketplace is not uncorrelated to their acceptance criteria. (E.g. University of Phoenix.)
[1]http://www.collegexpress.com/lists/list/colleges-that-do-not...