This will hurt banks with direct dealings in the US, but I think most smaller banks will just set up a shell bank to handle their US-based transactions. The shell bank doesn’t have any US depositors and the actual bank doesn’t care what US regulations are.
The effect will be an expansion of underground banks, exactly opposite what Obama wants.
You don't have the whole story - every bank is required to certify that they either:
1. have no us account holders, or
2. any accounts of a us person, or where a us person has control, are reported and are not used to avoid taxation.
The "Shell" bank would be required to cease transacting with the actual bank.
To avoid the additional compliance problems - many smaller banks just outright refuse US Customers, in that way they can certify under rule 1. Most others force US Customers to sign draconian release documents which basically allow them to give my account information to any one they desire, at any time, for any reason AND, if any damages occur due to that release, I am the one that is responsible!