In the nineties someone survived a season on as low as around 1700€ budget. But I guess it will take at least some 3000€ these days?
It is better reserve in advance – might be tough to find a long time stay if you don’t know anyone in the village etc.
One good place for looking accommodation and prices is http://www.interhome.com.

Also, call/mail to the Chamonix tourist center and ask for Chamonix Sud contact information – Chamonix Sud (a city block near Aguille du Midi down station) is cheap but the flats are also in pretty awful condition…a price you must pay for low budget accommodation.

St. Anton isn’t that familiar to us (only visited it briefly in the early 1990s). But one good advice in general is trying to hook up with some older “ski bums” and try find a place to stay from someone who’s been there and know all the essential contacts. Look for internet discussion forums, ski events and trade shows etc. And don’t be too shy to ask – usually there are some people that try to find flat mates for the winter (for lowering the overall cost).

Keep looking out blog. We have some interesting interviews coming that might give you some more insight (we are going to interview some people who have spent loooong times in Chamonix etc.)

Good luck to you, go for it – You won’t regret a season spent in the Alps!