The answer may surprise you but it's Yes, banks invest heavily in bitcoin.
Here is a non-exhaustive list of banks that invested in bitcoin in 2017: Goldman Sachs, JP Morgan, UBS, BNP Paribas, Societe Generale, CityBank, Barclay's.
By investing in bitcoin, I mean either bought directly from bitcoin or established partnerships with bitcoin and blockchain ecosystem companies.
In particular, 2017 saw futures trading on bitcoin (more info) on the Chicago Stock Exchange, where investment funds and banks rushed.
Why are banks investing in bitcoin? Simply because they understood that it had become a phenomenon too big to ignore and that bitcoin would take a significant place on global finance.
For the anecdote, JP Morgan boss Jamie Dimon publicly stated in October 2017 that bitcoin was "stupid", a scam and would fire any of its employees who traded bitcoin. The impact of the case was global, relayed in the media and bitcoin immediately dropped by 20%.
What did JP Morgan do in the wake? The same day, the bank took the opportunity to buy several million dollars of bitcoin. It was only a manipulation of prices to buy bitcoin, which proves that he did not think a word about what he said about bitcoin. Funny, no!