Could a solar flare shut down banking?

A severe solar storm could disrupt banking, but it would be more accurate to say it could cause widespread outages and delays rather than completely shut down the entire global banking system.

The main risk is not that a flare erases bank balances. The bigger issue is that a strong geomagnetic storm can disrupt infrastructure that banking depends on:

• Power grids: a severe storm can contribute to major power outages, which could knock out ATMs, card readers, branches, and data centers.
• Communications: banking depends on telecom networks and data links, and those can be degraded during a major space-weather event.
• Timing systems: some financial systems rely on precise timing sources, so if those are disrupted, certain trading or payment systems could be slowed or temporarily halted.

Your account data would usually still be protected by backups and replicated systems, so balances would not simply disappear. The more likely effect would be loss of access and transaction delays until power and communications are restored.

So the short answer is: yes, a very large solar storm could severely disrupt banking, but a total permanent shutdown is unlikely. The exact impact would depend on the severity of the storm and how prepared institutions and grid operators are.