Preventing this kind of fraud is a solved problem. The reason it still happens is that banks are forced, through competition, to minimise "identity proving" burden for consumers, in a "get credit now with instant approval!" kind of way.
At the moment we're stuck in a "marketing armageddon" of banks competing with each other by not properly verifying identity before granting credit or transferring away money. This seems to me like a Tragedy of the Commons.
If, across the board, people were required to prove their identity properly before banks rely on them, then the problem would go away overnight. It'd be a bit more tedious for consumers, but I don't see how that would cause banks to fail. The cost would merely move from fraud to identity verification.
Perhaps some people wouldn't be sold credit that they can't afford, but I don't buy that such people are keeping the banks afloat. Before banks stop operating retail banking services, I'm sure they'll just start actually verifying identity properly to keep that market.