No, I am quite certain that banks have better things to do. Like building their own version of Ripple to attempt to stay relevant. BUT even if they did run nodes, it wouldn't make a difference. The Onion Routing protocol used on the Lightning Network doesn't care who is running a node; it will use channels with good local connectivity to you, not those holding more funds. It would also be a liability to host a node with very many channels, because the more assets being hosted, the larger target it is for malicious attack. I suspect that exchanges will need to mitigate their risk by opening only a few channels to proxy nodes, who do the same, and so on, in order to dilute the honeypot.
RE: What is the Lightning Network?