There are several great hostels in the villages up-valley of Chamonix, like Gite le Belvedere in Argentiere. I had a pretty positive experience there for a couple months last summer, although the owner was trying to sell the place at the time so I'm not totally sure it's still running. The clientele in summer was mostly climbers and hikers; in March it'll probably be mostly skiers (it had a pretty big ski storage area). The transportation to and fro Cham and the Aiguille du Midi telepherique is great during the day with a couple of caveats: one, there isn't an early enough train for catching the first car, and two, you can't get back to Argentiere except by hitchhiking after a late night in town. I, uh, found that out the hard way. Of course it's only 10k back to bed, so you can run or walk...
If you're staying for a while (e.g., more than a couple weeks), you can negotiate a lower rate. There is a 50 cent or 1 euro tax per day for public transport at all places you might stay; it's actually extremely useful, since you can ride all the trains and buses for "free" with a card they give you. With the public transport tax and kitchen use fees, a reasonable rate is 13 euros per night for an extended stay (that's what I got, at least). That worked out to cheaper than rent where I live, so I was pretty pleased.
If you want to stay for a month or more you can also consider seasonal lets of an apartment or a studio. There are several agencies in town, I think at least two of them run by British folks, e.g. http://chamserve.com/EN_Seasonal_Lets_Chamonix.htm (had a good experience with them last year, friendly people, negotiate your own deal); Or check the billboards in the hostels, there are always postings for apartments for rent or roommates wanted. Rates for a studio in town start I think somewhere around 450 a month (sometimes incl. utilities) but I think you'd need to rent for a few months to get their low rate. They even have apartments with bunk beds that you could share with others.
//Edit: Correction - March is still winter season during which rates are usually double, so you're probably best off with hostels or looking for room mates wanted postings.