Banks using blockchain technology will raise awareness of cryptos, might bring more users into bitcoin, but not enough to move the prices. If banks use THE bitcoin blockchain, it will affect bitcoin price a lot, since they need to buy bitcoin to send transactions to the blockchain.
Using blockchain doesn't necessarily mean buying Bitcoin. The blockchain can be serve as trustless record ledger.
It doesn't mean buying Bitcoin directly, but you are using Bitcoin. If your infraestructure uses the blockchain, then it's using Bitcoin even if you think you don't directly. These miners are getting paid processing every transaction your bitcoin-blockchain based business makes, so you better hope that Bitcoin is healthy. This gives entrepreneurs using the blockchain an incentive to directly invest in BTC as well.