(i think) it depends on whether or not the school is need-blind. colby and bates are not need-blind, but bowdoin and williams are, as is amherst. if you’re applying to a need-blind school, your ability to pay is not considered. however, if you’re applying to a need-aware or need-sensitive school, your ability to pay is factored in. i’m not sure if anyone knows how much it matters, all anyone knows is that it does.
if you think you won’t receive any aid (or only very little), look into the liberal arts colleges that grant merit scholarships. the midwestern LACs like grinnell, kenyon, oberlin, and macalester are examples.