Marine ships and boats have ports. Airplanes have airports. In scifi spaceships can have space ports. What would be the airship equivalent in a steampunk world where marine ships are still heavily used? So that avoiding the use of port would keep things non-ambiguous.
I like aerodrome.
In the early days of aviation, when there were no paved runways and all landing fields were grass, a typical airfield might permit takeoffs and landings in only a couple of directions, much like today's airports, whereas an aerodrome was distinguished, by virtue of its much greater size, by its ability to handle landings and take offs in any direction. The ability to always take off and land directly into the wind, regardless of the wind's direction, was an important advantage in the earliest days of aviation when an airplane's performance in a crosswind takeoff or landing might be poor or even dangerous. The development of differential braking in aircraft, improved aircraft performance, utilization of paved runways, and the fact that a circular aerodrome required much more space than did the "L" or triangle shaped airfield, eventually made the early aerodromes obsolete.
Airport implies a runway, as noted. A blimp does not need a runway and early airplanes might need a variety of takeoff directions according to wind. A large field suitable for hosting all manner of airships could be (and was!) called an aerodrome.
It's called an Airship Docking Station.
The top of the Empire State Building was designed as the mooring mast for an airship docking station for Zepplins, but it was never used. (The real reason was to make it taller than the Chrysler Building.)
In his 1908 novel "The War in the Air" H. G. Wells refers to them as "aeronautic parks".
As it turns out there are many kinds of airships in real life. The most common are airplanes, then helicopters, and there are still a few dirigibles running around (for steampunk, I assume these are the airships you care about). They come to dock and stage from locations that are generally called... drum-roll please... AIRPORTS!!!
Heck the name going around for places for spacecraft to stage from is "spaceport". Maybe not all that exotic - use some of the other excellent answers in this thread for that. However, the actual, factual answer to the question is the mundane airport.