This just reflects my experience and it's limited. I hope it's still helpful to someone:
You can get grand taxis or the bus (line 16). I don't think it's possible to get a "petit taxi" / red taxi to the airport. But I don't have enough knowledge to say this beyond doubt. The grand taxis are white vans, and you can group with other passengers. I am not sure how common and easy this is though.
I used line 16, but I was lucky. This runs every hour up until 22h00, according to the airport information desk. They didn't provide any schedules though. This line runs between the Airport and the main train stations in Fes. It might be unusual that foreigners choose the bus. The man at the information desk seemed a bit surprised with my question. Besides, I didn't see any other recognizable foreigners travelling by bus.
There are no identifiable bus stops either in the airport or the train station. The departure in the airport is in a roundabout on the right side (when you exit the airport). After the parking lot. (Please confirm with the information desk at the airport though. Google maps identifies a different departure location)
Airport:
The departure from the train station is near the roundabout, after the trains station plaza, to the right, when exiting the station.
I was lucky enough that I happened to see the bus and ran for it. Otherwise, I would have to go by taxi.
The current fare is of 4Dirham.
All locals I asked sent me to the taxis saying this would be preferable (I asked twice).
Bus is faster than I expected, taking approximately 40 min. It's a bit chaotic and quite full though.
Other than that, it's a wonderful way to experience the local culture.