Both. Banks may be interested in blockchain technology to expand their services and they are interested in BTC, too. First thing first, they are interested in blockchain technology, but their investment is still low in this field. Let us say, they are analysing it, but rarely develop products with it. They are interested in BTC for other reason. The name of the reason is: ransomware. Yes, banks are often targeted with ransomware. They have started to stack up with BTC just in case. As far as I know, some altcoin platforms promised dedicated products to financial institutions, but none for BTC I am aware of. Maybe because BTC is supposed to support a financial system of people instead of banks...
Yes, and I think some banks are now into bitcoin. Although not fully. They are accepting to exchange Bitcoins to Fiat which think is an indication that little by little. They are starting to accept Crypto-Currency. Especially now that we are in a modernized world. All our transactions are tend to place online since we are now busy with the other things. They are not yet fully accepting it because they are analyzing it first (as stated above) but soon and very soon as they see the potential of this coin. They may use it in their transactions as well.
You should not forget about the so called anonimity of bitcoin system (most likely pseudo-anonimity). This is what banks don't want, because of the AML and KYC things. Banks are heavily regulated, they need to prove that they don't let terrorists, etc. to use their system to arrange the financial part of their business.
Banks can always prove that they know the sender and the receiver of the amount of money going thru their system. (OK, maybe only the sender, but because you can only send money from bank account to another bank account, the receiver is also known by the other bank, so everybody is authenticated somewhere.)
Banks are only interested in the technology, as stated above, because they're using old GIRO/ledger systems (national and international). They're old, slow and difficult. Blockchain technology can help them speed up their processes, but because of the heavy regulations, they will only implement their own system based on the blockchain technology. World is changing, processes are speeding up and banks need to catch up, this is why they're investing in the development of this kind of systems. Banks are hiring blockchain experts, but not in order to accept bitcoin but to improve their speed. Bitcoin (and altcoins, and any kind of ledger system) is a competitor now to banks because of the easy and speedy processes, and banks don't want to lose their customers because then they'll lose a part of their profit. To protect their profit, they need to invest in a system, that help them stay on the market.