The bottom edges of my basement door frame is "spongy" and rotting away. I wonder if it is termites, or simply that the wood used by the builder wasn't treated, and the wood is rotting away from moisture, etc. Any thoughts?

Remove or break apart the suspect piece of rotted wood, break it apart and inspect the interior. Termites can often be seen in an active infestation. They create distinctive channels through the softer parts of the wood excavating open runways. Termite affected wood often looks like a shell with the outside layer and a few pieces inside being intact.

Rotted wood on the other hand is often wet, with white fungal growths visible in the wood. The compromized wood is not hollow, but is soft and spongy or powdery. A distictive musty / fungal odor is usually present. If termites are at home when you break into the interior of the wood, call a pro. They will be in adjoining wood and walls. From your description, I suspect wet or dry rot.