As someone that has gone through the process at Princeton, I will re-emphasize that there are no athletic scholarships. There is, however, very generous financial aid. With tuition etc. over $40,000, you would be surprised at the incomes at which you qualify for sizable financial aid.
For example, if your family--after tax--makes less than about $50,000 (ie: one parent is a teacher), you get a "full ride". I had a non-athlete friend at school in that situation.
If your family makes around $150k, you could get up to about a "1/2 ride". This of course depends on your much you and your parents have saved, invested etc., how many siblings you have in college and a whole host of other factors Someone making 100k/year with no savings and two other kids in college is very different from 100k/year with a million in the bank. Most Ivy League school websites have basic financial aid calculators on their webpage, so you can try for yourself if you don't believe me.
I know there is much misconception about people getting athletic scholarship from ivies, but often that arises because for people receiving them it's easier to say "yeah, I got an athletic scholarship" than explain that in reality you got in because you were good at your sport and then got a financial aid package commensurate with your parents income and savings.