Last night I turned the humidifier on for the first time on my new heating system (an entire new gas furnace and air conditioner was installed this summer and this is the first time I needed the humidifier).

To do this, I simply turned the knob mounted on the return duct on my unit to about 40% humidity.

This morning, I awoke and my digital thermostat was dead. I went down to re-set the breaker (it wasn't tripped but I reset it anyway) and there was water on the floor of my basement. It appeared to be originating from the furnace area, but I can't be sure. There was a light rain last night as well. I don't see evidence of flooding from outside, and I wouldn't expect to.

I have reset the breakers and flipped the cut-off switch on and off several times and the thermostat is still dead.

Is it possible that the humidifier caused the flood?
And if so, is it possible that my unit shorted out somehow?

It is possible. Call the company that installed it because it should be under warranty.

Just had my 2 heaters checked. Found that one of the Humidifiers was not working at all. The 2nd unit was working as the "filter" was very clogged from "calcium and magnisium" which is due to the very hard water in our area.
Previous Heater repair service, had advised me that one of the electrical solinoids that controlled the water dripping onto the filter was not working.
He replaced (or told me he replaced the solinoid - cost $125 just for the solinoid) and all was working. With the service today, I got up into the attic space above our garage (where the heaters are located in most homes out here in the desert in AZ) and when serviceman opened the cover for the humidifer, the screen was "very clean", and no water was dripping on the screen when Humidifier was turned on to 40%. Further investigation showed that the solinoid was never changed in the first place, and we were "scammed" in paying for a solinoid that was not needed. The 2nd heater Humidifier screen was very clogged with calcium and magnisium, which indicated that the humidifier was working properly.
For your situation, I would definitely have the serviceman return to inspect the Humidifier solinoid, to see if it is working properly. If this is not working then the water will continue to drip onto the screen when you turn on the Humidifier to 40%. The water will stop dripping, when the temp set in your house is reached. Then it will run water again when your Heater turns on for more heat demand. Sure sounds to me like the solinoid is not working.

With this service today, I found out that