First let's assume it uses no money! How does it do that?
For one, every customer gets a debit card. Then, go back to the origin of banks. They started as a place to store gold. On mars, account holders would deposit pure metals; Gold, Silver, Copper, Aluminum, Iron, etc. Using their debit card, customers would spend those metal deposits just like money.
Second, the banks do not make loans! The customers make the loans. How does that work?
A person needing a loan goes to the banks website and puts in a loan request (I would like 10 kg of platinum please.) Public comments would attach to each loan request (Q&A basically.) Bank account holders would then put in bids on that loan (I offer 100 grams for 0.1% a day.) At some point they reach 10kg or no loan is made. The borrower can wait for better offers to continue to come in or accept the terms of those that have come in already (only the best of those up to the 10kg are summarized for the borrower so they can decide). The bank handles the transaction for a small set fee (not a percentage of the loan. Other banks will compete by offering lower fees.) This is the first way banks will make an income. The second way is as a commodity exchange offering one metal for another using a simple bid/ask system. There is no fractional banking. The only audit needed is to see what mass of metals they hold which they will make freely available to all their account holders.
Easy Peasy. Anybody with a decent reputation can run their own bank. Caveat Emptor.
Could they handle currency from earth as well? Sure. Why not? It's handled exactly the same way as the metal deposits including a currency and metals exchange. The only regulation required is that account holders have complete transparency to bank holdings. It's all just contract law.
What does the borrower do with his 10 kg of platinum? Most likely he deposits it in that same bank and uses his debit card to conduct his business. It's all just an electronic transaction really except the platinum is for real and held by the bank.
This suggests another mars occupation. Someone to certify the purity of metals for a fee. A person's name and reputation will come to mean a lot on mars. A person's word and even the lowly handshake could return to it's former value. Politics would only exist on a very personal level.
Note how unlikely a bank run would happen if all account holders knew their deposits existed and were secure. Even if there were a bank run; No fractional banking means the bank continues to be in business without danger even if they do end up with fewer deposits on account.
They may also operate tradition insurance operations or be prevented from doing so; it's all contracted.
One more thing to realize. This is not complicated. Getting loans will be easy because everybody on mars will be a land owner and can use that for collateral which the bank will hold against repayment of the loan. Land will have a known mineral content. Everything is transparent. Easy loans means a business boom. Liquidity is no problem.
Update: Crowd sourcing.
Note how unlikely a bank run would happen if all account holders knew their deposits existed and were secure. Even if there were a bank run; No fractional banking means the bank continues to be in business without danger even if they do end up with fewer deposits on account.
They may also operate tradition insurance operations or be prevented from doing so; it's all contracted.
One more thing to realize. This is not complicated. Getting loans will be easy because everybody on mars will be a land owner and can use that for collateral which the bank will hold against repayment of the loan. Land will have a known mineral content. Everything is transparent. Easy loans means a business boom. Liquidity is no problem.
Update: Crowd sourcing.