Color if no communication

I’m using the Beam over the HTTP API.
Is there a way I could set a color if the wifi communication drops?

No, because the only way to change the colour on a device is via wifi. If wifi drops, so does your ability to change the colour.

so it will just stick with the last communicated settings even after power cycling?

Correct. A standard power cycle ( single off then on) will retain the settings. A quick double power off on most bulbs will reset to a neutral white. And 5 power on/off cycles will reset the device completely.