Last December I met in Zermatt (by many regarded as one of the best ski resorts on the planet) a 19 years old boy from Tel Aviv.
He was very well educated and friendly and we started chatting on the lift: after a bit he explained that he was admittedly money-less as he was studying Mathematics at University and was getting ready to serve in the Military in the coming year.
He added that he loved skiing and normally he was able to ski only in Bulgaria as that country had the cheapest ski resorts reachable from Israel.
I looked at him a bit surprised, given that we were chatting in one of the most expensive ski resorts in Switzerland.
He then explained that this December he planned in advance and with a friend he was able to book a Ryanair ticket from Tel Aviv to Bergamo (close to Milan) for 10 Euro only.
Once in Bergamo, he booked a seat on a very cheap bus to Aosta for just a couple of Euros. In Aosta (north west of Italy), he booked a cheap hostel there and on a daily basis moved to Cervinia by bus where he was able to ski for less than 50 Euro a day.
In low season (before Christmas) you can make the most of your ski days as the slopes in Europe are basically empty.
From Cervinia (Italy) you can ski and cross the border with Switzerland (crossing the Matterhorn Glacier, right in front of the majestic Matterhorn) and reach even Zermatt, if you want to do so.
This is just an example that even if skiing requires dedicated equipment, travelling, ski-passes, etc., if you are passionate for this sport, with a bit of planning you can do it also on a budget.