It is very difficult to answer since there are hundreds or thousands skis resorts in the Alps.
But to narrow down the potential candidates, I would focus on very small ski resorts in Austria.
The reason is quite simple, smaller resorts are generally cheaper than big resorts and Austria is generally way cheaper than Switzerland, and cheaper (at least concerning ski resorts) than France.
If had to recommend a single location, it would be Laterns in Vorarlberg, Austria.
The first sentence of the homepage says that it is one of the cheapest ski resorts in all of Austria.
This ski resort also offers a good opportunity called Vormittagskarten mit Rückvergütung .
This means you can pay for a ticket in the morning and then as soon as you've had enough snowboarding you can give it back
and then you will receive some money back.
As a beginner this might be a good choice.
Laterns is not the only ski resort that offer such tickets.
Nowadays it is quite common in the Alps (or at least Austrian Alps).
A lot of ski resorts also offer perks if you're a student or in military service.
So don't forget your student card if you have one.
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Just one thought to start off.
If you are a beginner, bad slopes can be a disadvantage ...
I would have a look at Germany, and more particularly the Allgäu region .
You have some smaller resorts over there, which are a bit cheaper than resorts in Austria.
In the French Alps, places like Chamrousse and Saint-François-Longchamp are not too expensive.
The last one is a very nice place for beginners.
Look for a flat and do some self-catering to beat down the price.
Northern Italy, with the exception of South Tyrol (Alto Adige), should not be too expensive either.
Here I am thinking about Lombardia but also the Trentino.
If you want something really cheap, I would not go to the Alps.
Czech Republic or the Tatra mountains could be a good choice.
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I can recommend Livigno in northern Italy.
It is a tax free zone so everything is a bit cheaper than in the rest of Italy.
It is close to the swiss border.
The skiing is probably boring for the advanced but there are lots of smaller slopes for the beginner.
They also have a sleigh track which is fun and easy regardless of your skiing skills.
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Arguably cheapest ski resort in alps are in Slovenia.
Kranjska gora or Vogel do not offer so much than others in austria, but still if you a beginner it Will be just ok.
Of course for 30-40% less money: Info about Kranjska gora
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Switzerland is definitely more expensive but I am not sure Austria is necessarily much cheaper than France.
It's going to depend a lot on the resort and there are many small and even quite a few large but less famous resorts in France where accommodation and the like are somewhat cheaper.
I don't know their ski resorts but you could consider Italy, Germany or Slovenia as well.
To reduce cost, you need to consider a few things:
Prices vary a lot, it's more important to see what you can find than settle on a country or region from the get go.
Lesser known villages at lower altitude and accommodation
that's a bit far from the lift/slopes
are usually markedly cheaper, even in the same area.
And beware of spending more on transport just to save a few euros. Restaurants, services and even groceries are usually expensive in resorts, go for self-catering and shop in the valley to save money (that's easier with a car obviously).
Don't worry too much about ski pass prices.
Even large resorts offer cheaper “beginner's passes” that only cover a few slopes or even free lifts and this should be more than adequate for you (but you do need to check if there is something like that, it does not exist everywhere).
Don't buy an expensive weekly pass either, you are probably going to get tired quickly or want a day off from skying (bad weather is also a possibility).
Just buy passes for a few hours or half a day when you feel like skying.
Renting equipment is also going to be an important part of your budget
but I don't know any shortcuts there.
Also, you asked about the Alps and they certainly offer some very good resorts and breathtaking landscapes but as an absolute beginner, other lesser known areas (Vosges, Pyrenees, Massif Central, Jura, Black Forest, Ore Mountains, Tatras…) could be cheaper and easier to reach for you.
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Several resorts offer skipass for free (usually during the beginning and last weeks of the seasons) if you book accomodation with them.
Depending on the weather / luck that can be a great (or lackluster) deal.
Here is the offer from Livigno https://www.livigno.eu/en/skipass-free Offer from Val di Sole: http://www.valdisole.net/EN/Free-Ski-Weeks/
Previously I have seen such offers form Bromio http://www.bormioski.eu/ ,Madonna di Campiglio
http://www.campigliodolomiti.it and Les Arcs http://www.lesarcs.com but can't find anything about that on their webpages currently.
Maybe they will publish that later in the season.
You can also write them and ask if there will be such offers in spring 2017.
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I'm a total beginner. Where should I go if I want to start skiing at the minimum price? Don't worry if the slopes are bad/small/too easy, if there is snow and it's cheap, that's good.  I am only interested in Alps as I am going to travel through the French Alps and Switzerland, and perhaps some part of Northern Italy.
Which is the cheapest ski resort in the Alps?
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Small resorts are usually cheaper than large ones, and Austria is generally cheaper than France and Switzerland. Laterns  in Vorarlberg, Austria is a good choice. In the French Alps, places like Chamrousse and Saint-François-Longchamp are not too expensive. Northern Italy shouldn't be too expensive, with the exception of South Tyrol (Alto Adige); Livigno is a good resort to try. Slovenia is probably the cheapest in the Alps, although you could find other affordable resorts with beautiful scenery in lesser known areas (Vosges, Pyrenees, Massif Central, Jura, Black Forest, Ore Mountains, Tatras…). Austria is considered a good destination for beginner skiers that is not too expensive. However, France, Italy, Germany and Slovenia may have something to offer. Away from the Alps, the Czech Republic, Pyrenees, Vosges Mountains and Tatra Mountains may be a good choice.
Austria is a good place for ski beginners and not too expensive compared to other places in the Alps. Ski destinations in the Alps include France, Italy, Germany and Slovenia. Other areas are the Czech Republic and Tatra Mountains. Good ski areas away from the Alps may be good for beginners and cheaper. Examples include the Pyrenees and Vosges.