The people will probably have american currency on them at the time of the move and continue to use it for awhile, Midnighter is actually.. ridiculous and idealistic once you get past the 90's gritty attitude, and he'd be more interested in a barter system or some way of living without currency. Basically, everything they need to survive is free, and they can trade with each other if they want anything else.

There is definitely not going to be trading going on with Earth for anything other than what the private citizens go out of their way for (ie, if they feel like using a Door to get back down to the City to purchase things for themselves with their own money), but yeah. They never went into details in canon, but they took on an entire nation of refugees in the Authority run and obviously had some way of generating or growing food on-board, because they didn't deal with anyone outside in order to sustain the population.