I'll give you a "makes sense" answer rather than an authoritative one, since I've read the wiki and watched the show but haven't read the books:
Basically the answer is taxes of various kinds, which translate also into indirect trade as well.
So, there's no separate currency for the North - all of Westeros uses Gold Dragons, Silver stags and so on; so the Starks don't issue their own money. That means the Starks don't make their own money, like the crown can.
... except that's not quite true, since we're not talking about debt-based bank-invented money like we have today (yes, most money today is created out of thin air by banks when they lend out money, except for a fraction which they must actually have; look it up or watch this interview) - but rather metal-backed, or actually, metal-proper money. So the money's exchange value corresponds to a great extent to the amount and kind of the precious metal in it. I'm not sure how much GRRM has fleshed this out, but the point is that even if you don't have your own mint (and customary coinage which is accepted by many), you can still get money by arranging for minting of coin in exchange for the raw metal. The crown will probably take some fraction for itself, but that's not very far from making your own money.
Now, it's not clear to me that the Starks do much of that, since the metal-rich area of the 7 Kingdoms are the Westerlands, the Lannisters' domain - and that's where they get their money mostly. Still, since the North is so large, there's bound to be some kind of mining of precious metals going on.
Which brings us to taxation. The crown can collect taxes in its own money, which is really neat, because instead of you having to chase people and take their stuff, they know they'll be in trouble if they don't get you (= the crown, which has the mint) to give them money, for them to pay you as the tax. Why would you give them money? Because they provided goods or services of some kind. So they'll essentially come asking to service your needs... ain't sovereignity great? You can really mess with'em :-) Now, I don't know if that works on the sub-kingdom level (i.e. Crownlands vs the Reach or vs the North), but it very likely does to some extent, since the other sub-kingdoms need the crown's money to run the money-based part of their own economies. So in exchange for whatever services and goods the North renders the crown, it gets money (and probably goods produced in the South); it may then need to also pay some of that money back as tax.
Internally, a (sub-)kingdom like the North is more likely to tax in kind - take usable goods or require rendering services. The requirement is either directly from individuals or by taxing lords of different regions/castles/towns etc; or maybe both. I guess the taxes are probably partly in crown coin and partly in goods/services, since both are necessary to the Starks in different contexts. There might also be taxes on individuals in cities, or on roads, and those are more likely to be monetary. Thinking about the character of the Stark rule I would guess that's less likely, and maybe houses like, say, the Manderleys and others get money through taxation of individuals, or of commercial activities, the the Starks get more money, as opposed to goods, from these houses.
Sorry for this being a bit vague and inspecific, but likely I said, it's what is almost certainly the case upto some details.