Btukfyl

Has any story, movie, TV show depicted a spaceport on either of Mars' moons, rather than a spaceport on the surface of Mars?

In Morning Star, the third book in Pierce Brown's Red Rising trilogy, Phobos is a transfer point for shipping helium-3, with 30 million inhabitants:

The barren rock of Phobos has been carved hollow by man and wreathed
with metal. With a radius of only twelve kilometers at its widest,
the moon is ringed by two huge dockyards, which run perpendicular to
each other. They're dark metal with white glyphs and blinking red
lights for docking ships. They slither with the movement of magnetic
trams and cargo vessels. Beneath the dockyards, and at times rising
around them in the form of spiked towers, is the Hive - a jigsaw city
formed not by neoclassical Gold ideals, but by raw economics without
the confines of gravity. Six centuries' worth of buildings perforate
Phobos. It is the largest pincushion man has ever built.

Transshipping helium-3 up from Mars and onward to the rest of the solar system is Phobos' purpose:

In the aftermath of my escape, the Jackal initiated an immediate
moratorium on all flights leaving Mars for orbit.... Ultimately, not
even the ArchGovernor of Mars could ground all commerce for long, and
so his moratorium was short-lived. Billions of credits lost every
minute the helium-3 did not flow [via Phobos].

One of the earliest (1951) attempts at hard SF about Mars (that was also well-written), Arthur C. Clarke's Sands of Mars features a spaceport on Deimos. The idea has been used many times since then -- the exquisite Poul Anderson story "The Martian Crown Jewels" for instance. Phobos and/or Deimos have also been alien spaceships a number of times, but that doesn't count, I guess...

Yes. The Doom game was set on Phobos and there was a spaceport.

Phobos is the larger and innermost of the two moons of the planet Mars, the second being Deimos. It is the scene of the first Doom episode, Knee-Deep in the Dead. "Phobos" is the name of a god in Greek mythology and it can be translated as "panic fear", "flight" or "battlefield rout".

In Doom, Phobos is depicted with Earth-like gravity, a thick atmosphere, and having tall, seemingly vegetation-covered mountains; the sky texture for the episode was derived from a photograph taken of Yangshuo Cavern in China.

In reality, Phobos is a rock 22 kilometers in diameter with gravity less than a thousandth of that on Earth, and no atmosphere (even if an atmosphere could be generated artificially, the gravity would be insufficient to hold it in place). Phobos' gravity is so weak that a human being could escape it by jumping. In order to be more plausible, Doom 3 moved the plot to Mars.