Sort of like how we do now haha. As long as the fighting isn't at your doorstep you live your life as normal as you can. But for sure they affect the government and governance of the cities, Atlathka is extremely vested in annexing Kaldea at this point in the timeline and it would be the subject of most of their gatherings and the considerations of possible future action would be pervasive in other meetings as well.
Kaldea is no different as you will find in our latest quest. Even while talking about unrelated matters at hand the conversation will usually devolve into bickering between opposite ideals in how to address the looming Atlathkan threat. There are people in Kaldea who want to go into active warfare and others who want to cede the city to Atlathka in a peaceful process.
Atlathka is suffering most from this at the moment and is a large part of what started the war. Kaldea were charging tariffs on EVERY SINGLE VESSEL passing through the Kaldea Rock Channel, which is a pretty narrow passage. Ships that haven't paid tariffs aren't turned away... but tend (with an alarming 100% of the time reliability) to get attacked by pirates.
Currently, Kaldea is still letting ships through to Atlathka so long as they aren't carrying arms, armour, or magical implements that could be used for warfare... so they are watching for siege equipment and such. Atlathka is blockading land access though which makes up a TON of Kaldea's trade, Kaldea is the first land access to the Western Advance and the Broken Frontier range, all the villages and small towns on the coast etc. Atlathka hasn't been able to control the passage north as well as the western advance (which actually heads eastwards from Kaldea!)
Banks are very secure institutions back then. They are well guarded and their reputation was completely reliant on how they treated their patrons... if a bank was screwing people over, people wouldn't use it. So they were exceptional institutions in the real world and VERY secure.
The Kaldean bank is managed by its own institution that keeps out of the senate through the mercantile coalition and the aristocracy have ties to it.
They handle investments, loans, lines of credit and such. A lot of trade in the city is handled with banknotes between the large merchant houses. There are also forms of scrip in the city which have little to no monetary value but have a trade value and the bank will accept script as well as almost a form of stocks trading.
The bank also records and keeps track of the value of goods and resources in the city, posting updating and posting the going market rate for goods. This vital service means the bank has a lot of power in the city in general.