When you say "change the digital output to zero state, permanently?" you mean the Power-up State right?
If you goto MAX->Devices and Interfaces->NI PCI-6251->Properties->Power-up States and change the settings, are the settings not "sticking" after a reboot or something like that? The Power-up state should be configurable from MAX.
If all else fales, you could re-work the wiring going to your relay. Connect the +5V (from your card, should be several pins that supply +5) to the positive of your relay, and then connect the digital output to the negative of your relay. When your device turns on if it defaults to "high" on your digital output then you should have +5VDC on the positive and +5VDC on the negative (0VDC across your relay, which should prevent it from actuating). Then when your toggle the digital output to OFF, the pin will sink current and cause your relay to turn on (because it will be like attaching GND to the negative of your relay).
Be mindful of all things safety related no matter what you do.
