They're most likely still running all their transaction processing on a single mainframe (+another one for redundancy) with something like IBM CICS [1, 2]. And they most likely also have a lot more transactions than in the 60s (card payments, internet banking, etc.).
[1] http://www.ibm.com/software/htp/cics/ [2] https://en.wikipedia.org/wiki/CICS - yes, "Initial release: 1968; 47 years ago"