Like any business, banks want to be sure that nothing will prevent them from serving customers. Banks are dependent on the environment in which they operate and are exposed to external failures in this environment, but have developed contingency measures to overcome these effects. No, this is unnecessary and unwise. The money in your account at the end of 1999 will still be there at the beginning of 2000. Banks will not be issuing special guarantees because your usual consumer rights apply under your existing relationship with them.
Banks always allow for a wide range of risks to their businesses.
In the very unlikely event that something does go wrong over the Millennium weekend, Banks have in hand contingency plans designed to ensure that any effect on customers is minimal. Get in touch with your bank in the usual way - either telephone, or drop in to your branch.