You should not equate banks with the modern financial sector... there are huge differences. I mean the classic medieval form of banking. Which is pretty much a communication job. You give somebody interest for him giving you his money. And you invest this money to pay this interest and some revenue for you.
No need for fast communications, it worked when most mail needed months to arrive.
But for example concepts like shadow accounts (Many modern simple transactions need them) or ship funds are impossible without banks backing them up.
Please dont get the impression that I think banks are inherently worthless, they do perform a useful function that would be extremely important in space infrastructure. Real entrepreneurism seems pretty much dead here on Earth for a variety of reasons, but space economies will need to run at levels we would consider red-hot (practically glowing :lol

in order to work. Internally, the cost of labour on a base is going to be through the roof, and as such, any voids in the market will need to be (and will be) filled quite quickly. Because of that startup loans from a bank would be important in order to allow new arrivals to start a business quickly & get to work. That being said, that will likely only represent a colony after 10-20 years minimum. Noone is going to care about economics in the slightest until the base is stable engineering & lifesupport wise.
The most important fact about barter systems is, that it never worked. What will you do, when you need X from Y, but Y does not want anything that you can offer him? That was the point where currency was invented.
And with the currency came banking quickly.
Yes, but a currency system has to be enforced top-down for it to work, primarily because having a currency system is beneficial for local government in terms of stability & order. Individual colonies may eventually develop local currencies (again ~10-20 years there), but how will they exchange with regards to Earth based ones? If news of an accident, even minor, on the Phobos colony reaches Earth, does the value of a Phobos-dollar drop relative to USD?
It would seem to me that actually implementing this currency would be less effective than the barter system, because agreeing on that exchange rate would be so difficult, compared to just hashing out something mutually beneficial at the local restaurant/bar/whatever.
The only way I can see a currency system really forming would be around the first really big trade center, a location that sees a lot of regular traffic, & most importantly, has something that Earth wants
badly. That would be to me, the only way that off-planet colonies would really get enough leverage with the buyers on Earth, because the value of what they're sending back would be equal or greater than whats being sent out.
My best guess of how that would unfold would be an L4 or L5 space colony, maybe not as big as the Stanford Torus, but big enough to act as a way station for deliveries to/from the asteroid belt & Mars. The local management is sitting on top of the chokehold for all of those valuable resources that earth wants, (copper, nickel, other precious metals. Just look at the price of those things right now, then think of how valuable it will be in 50-100 years) and they finally have enough leverage with Earth to work out a stable exchange rate. Other space colonies adopt their currency system (the Lagrange Loonie?) in order to benefit from the influence it has, forming possibly the first off-planet power bloc.
A classic bank is no actor, it is an assistant. A classic bank doesn't own much, but acts in the interests of its customers. No comparable to today, where the bank owns you, when you give it your money.
Imagine X orders a few tons of hydrogen. Now, you should expect that X can pay the show. But how could you be sure? The only reliable way is a bank that confirms that X has deposited the money for the transaction and can clear it on delivery.
That would present an interesting problem, but my feeling would be that before the currency system I mentioned above is set up, a lot of space transactions will be pretty simple in practice. The nuclear freighter captain & his crew haul that LH2 ( not a great example because its a fuel, I would expect electronics, solar panels, a nuclear reactor...) to the customer, (again Phobos, Ceres, L4/L5, maybe even Mars surface for Zubrins methane fuel production plan) they get a month of R&R & spacecraft servicing free of charge, along with some compensation. I would figure that after spending months in space, most crewmembers would rather take at least part of their cut in a tangible way, since actually using that money is a long ways away.
Part of the problem would be how to structure infrastructure productively around launch windows. Having 18 months of quiet between mad dashes to depart at phobos station sounds like a recipe for eventual disaster.
A spacecraft like the DG would be a multi-hundred-million investment, regardless which currency you will later use. It is comparable to a single factory on Earth.
If you sum up all current satellites and estimate their current value, you would notice that we have the wealth of a middle-sized country in space today. Alone the ISS represents a few small countries.
At the moment, yes, but the situation would probably be a bit more complex. An actual interplanetary freighter like the Deepstar, Pegasus, or an IMS ship would be a huge investment for a huge return. More infrastructure should hopefully cheapen its cost over time, but its hard to imagine one being built without the capital of a major shipping company behind it. So, there goes that whole idea of rakishly independent adventurer captains flying their own ships wherever they please...
On the other hand, the goods that could be in high demand are not necessarily high mass or high volume, if they were delivered at the right time (electronics, high memory data, maybe even nuclear fuel rods if the captain is pretty gutsy). An independent captain might be able to wrangle together a very cheap, smallish freighter, with parts reminiscent of the LEM ascent stage, inflatable modules for living space, & no artificial gravity. If he/she were willing to make those supply runs, particularly high delta vee ones late or early in a tranfer window, they probably could get by.
The really interesting thing would be the Jovian system. If setting up a habitable station is possible at any point there, trade infrastructure could grow very quickly.