The biggest advantage of a ground plane can be seen with digital PCBs, when the ground plane provides the shortest possible path of ground return current for all signal or power supply currents. At the same time the ground plane offers the smallest area for all involved current loops and the least inductive paths for all the ground return currents.

In your application, where the ground plane isn't used for any ground return current (those, caused by the charging of stray capacitance excluded), all these advantages aren't available. Nevertheless, here a ground plane can also be of help due to its shielding abilities against electrical fields. This the more, the lower the inductance of its connection to signal ground of micro board is.

If a PCB is mounted like a wall of an enclosure, as in your application, where the keyboard looks through one enclosure wall, the ground plane can help to keep the Faraday cage closed, if a metal enclosure is used. In this case, the ground plane should be connected to the metal enclosure by the help of some 10...100nF capacitors. Keep the inductance of these connections as low as possible and distribute these capacitors among the whole ground plane, best, where the PCB is mounted to the enclosure by the help of screws.

Connecting the ground plane to enclosure, even if this done only via capacitors (for higher frequencies only), can tremendously help to keep the application immune against ESD events, hitting the keyboard-display-board!

So much to the ground plane.

We know that you have a mixed application using an analog section. Unfortunately, multiplexing the display in the way you intend to do, means routing the multiplexed currents along cables, can produce lots of noise and can highly degrade analog signal integrity: As the multiplexed currents and their associated return currents cannot be routed directly next to each other within the ribbon cable, because there are six different return current paths (ULN2003) instead of only one, enormous loop areas are created. So, the consequence is the generation of enormous high frequency electrical, magnetical and electromagnetical interference.

Unfortunately, due to the rather high display currents, adequate low pass filtering at output of 74LS541 is difficult. So, I would strongly recommend to avoid the routing of multiplexed currents along cables. Either you could move the multiplexing onto the display board, or you could use a static display. I would use six 74HC595 to directly drive the seven segment displays and would use a serial data transmission to them. You would only need three lines: Data, Clock and Storage. The whole transmission could be done in about 1msec.

Kai