Banking in Redmont is really just "leaving your money with a moneychanger and hoping he doesn't run off with it". There are obviously legalities in place to help in this matter, and permanent guards are on hand to stop such an eventuality, but leaving money anywhere but your own safe in your own house with your own locks and traps is never 100% safe.
The money changers in Redmont each have their own accounts, and one cannot withdraw or deposit at any other moneychanger. They do provide a service to change currencies over, at a 1% transaction fee, as well as make loans at a repayment level individual to each lender.
The best way to secure funds is by making your pile of money seem less easy to get to than the next person's pile of money. Carrying large sums of it in public is a stupid idea, but sometimes necessary (buying a house etc). In this case, arrangements can be made with the guards to escort you, or the buyer to arrange a pickup of funds at your residence.
Most common folk dont have enough money worth stealing- a few copper pieces, a few silver pieces- enough to pay for food and tax. Adventurers and merchants however often have no fixed residence and large amounts of cash on hand. These are the people most at risk, and therefore the ones assumed to have the most fighting skill. Shame that it is technically illegal to weild your weapon in public. Some believe that some theives are hired by the council to assault unwary traders and adventurers as a "flashy tax" (flashing their money).