Many bank servers and account databases are or can be hardened against EMP, solar flare. most also record data off line in read only form. So your money can com back after. notionally. Its why some banks still have dvd burners, vaults of flash chips. Or should.
The key is the "or should" some banks have no safe guards.
If EMP knocks out the grid and computer systems them banking will be 4th or 5th on the list.
Hospitals; medical life support and pharmaceutical refrigeration. Its why many hospitals have compressed air lines in the walls.
Food transport and refrigeration. Rail will be a priority and diesel trucks with torches or lamps. Repairing the electrical systems in a truck is easy; the wiring is generally heavy duty. EMP will only fry a replaceable chip or two. A lot of museum vehicles will be used. Water transport will matter. Remember every unemployed man woman and teen will be taught to replace automotive wiring or rebuild power plants and grid wiring. Most power and transport would be up in a month or two.
Printing facilities will be the third priority because going back to printing ration vouchers, accounting paperwork, Local Employment Trading Systems Though these can be done with pen and paper), emergency micro currencies and paper bitcoin wallets. Some people have Faraday caged their bitcoin accounts.
The Forth will be banking. Or traffic lights. Police can do their jobs; guns and badges are unaffected. They will be using bicycles, horses, repaired trucks as needed. Volunteers will be deputised.
I would not expect the rebuild to take more than three months and the majority of first world countries would get back to a running 1960's cheque book economy with working grid in a month or two.
The poorest in the third would would be virtually unaffected driving a power change. The Amish and Mennonites in the USA would be heroes.
It might be a good idea for someone to kick off a EMP survival club that pre trains people in all these things and encourages friends, families and co workers to back stuff up on a flash chip in a metal box or bag. Teaching people how to take the contents of your fridge and preserve it without electricity should be on the list.
Its survivable. Some would die but not as many as might be expected. You're back to 1960 not 1860.