Currently banks are not working with Bitcoins. Even if they start to work, they will most probably just make Blockchain transactions. Why they must use paper Bitcoins, if just some Bitcoin users use paper Bitcoins. Even Bitcoin use in bank sector is future, currently on one bank use Bitcoin.
It is very simple really
If Bitcoin banks should spring up, they will be earning through Bitcoin, obviously. Thereby, they will be vitally interested in the success of Bitcoin, right? But Bitcoin is often reproached for not being very useful for offline transactions (which is partially true). Paper bitcoins (or bitnotes) would greatly help in this respect since people would use them as any other paper money out there (say, dollars). The only drawback is that Bitcoin banks would have to withstand a great temptation to issue more bitnotes than they have real bitcoins in their "vaults" (i.e. wallets)
So who is going to issue these Bitcoin paper certificates? The banks themselves or people would be able to write themselves some certificates as well in the same fashion as we write checks today on a paper pad?
Bitcoin banks will issue them, allegedly in correspondence with the amount of real bitcoins they have