It was savanna during the Neolithic Subpluvial until about 3000 BC. The end of the wet phase caused Sahara nomads to settle down in the Nile valley and to start the early Egyptian civilization. But that was prehistory. It's always portrayed as a desert, because it has always been a desert in the recorded history.

The fertile crescent is still fertile and not a desert. It's composed of the Levant and Mesopotamia. The Levant has always been wet, but Mesopotamia has been mostly dry, with some wetter phases when there was a steppe, particularly in the northern Mesopotamia.