Random data failure indicates clocking errors which points to signal integrity issues. Is it clocking and latching the data remotely? Examine grounding, crosstalk, voltage integrity and consider foil or braid shield, interleaved grounds, R terminator or pull up/down on clock to match impedance of driver and ribbon cable ( 120~240 Ohm) to prevent ringing.
or... Design race conditions with data changing on clock edge...with ripple and latency.
If grounding one finger and holding the cable helps, it may be a common mode noise issue related to cross-talk. If the voltage drop is excessive on high current digit drivers, there may be crosstalk from multiplexing. When done and got it working, add polyurethane, silicone or hot glue for cable strain relief near soldered end connections. Same goes for large components like LEDs or big caps. Ensure Vcc is clean and digit driver voltage drop is constant regardless of number of segments active.
etc etc.
In any case it is a design problem.
Are the random digits dim or bright?
If it is a solder issue , it may be sensitive to finger pressure or small shock waves to board. (bump)