The most basic call center employee needs access to data of all the customers, since any of them may call. How can you partition the data and at the same time achieve seamless customer experience wherever the customer may contact you?
Partition the call centre employees according to the least significant digit or digits of the customer's telephone number. Employees A, B and C deal with customers with phone numbers ending in 0, and can only see records of those customers. Employees D, E and F deal with phone numbers ending in 1, and so on.
This is how it was done when I worked in the civil service nearly 20 years ago (well, there it was alphabetically by customer surname, but it's the same principle). That was done for logistical convenience, because we had huge quantities of paper. The records of any customer who might call me would be within 10 feet of where I was sitting, but it had the useful side effect of making it obvious if I went looking at records I wasn't supposed to.
Granted, this approach in itself wouldn't stop someone copying everything they can access onto a DVD, but if done properly, it would limit the number of customers who want to sue you. It would also give you a head start on figuring out which employee has gone rogue or wrote their password on a Post-It...