At Navy, it's the independent non-profit Naval Academy Athletic Assn (NAAA) that funds recruiting visits, coaching salaries, stadium improvements, builds athletic facilities, buys gear, pays for travel, runs revenue-producing events and ticket sales, owns logo rights, etc. No tax-payer or USNA budget funds. NAAA's mission is to promote and support NCAA Div I sports. They have no mission to recruit academically, except to work closely with USNA Admissions on potential candidates.
USNA Admissions probably has a small recruitment budget. They run NASS, oversee non-athletic CVWs, do recruiting presentations in under-represented areas, and so on.
The revenues raised by NAAA, including donations, ensure a consistent level of financial support for NCAA athletic programs, independent of govt funding.
I suspect USAFA has a similar arrangement, the Air Force Academy Athletic Corporation (AFAAC). You can see the similarities:
www.navysports.com
http://www.goairforcefalcons.com/ot/afa-athletic-corporation.html
The theories of why varsity athletes are attractive to service academies has been cussed and discussed on many threads here over the years. It's part of the whole candidate theory and the art and science of SA admissions.