I appreciate the effort AHA puts into competitions even though I don't plan to enter one anytime soon (if ever). I like to read the stories, read the recipes, etc., and I consider the recipes that come out of these competitions to be a great insight into current interpretations of beer styles. As these recipes accrue, they become historical records of styles "back in the day," as well as a snapshot of the evolving face of homebrewing.
So for me, to the question "how much time/resources does AHA put into competitions," the response would be "just enough." In addition, the AHA advocates for homebrewing, provides this forum, has an excellent magazine, and otherwise gives me a lot of bang for the cost of membership.
I don't care about the dollar amount (not even as a percentage) because it wouldn't make sense out of context of understanding the entire AHA budget, which would take more than numbers on a page.