The evidence suggests Kryptonians do need to breathe...
... some of the time, at least.
Restricting myself to evidence from the older movies only, in Superman: The Movie (1978), when Luthor shoves Superman into a swimming pool, with that chunk of Kryptonite hanging from his neck, we see Superman exhaling bubbles of air while he's underwater.
We then see him gasping for air when his head emerges from the water, and he appears short of breath, regularly pausing to take in more oxygen, while pleading with Teschmacher to help him save the people endangered by Luthor's missiles.
Granted, Superman is depowered by the Kryptonite in this scene, but then as far as we know, Kryptonians are also powerless while under a red sun, as they were on their home planet.
There are multiple indications of Superman needing to breathe, even with his powers, in Superman III. For example, when Superman splits into two beings during the junkyard fight, the Clark Kent half of him can be heard gasping for air while being enclosed by the car crusher.
And he subsequently defeats his other half via strangulation, a tactic which shouldn't be terribly effective against a being who doesn't require oxygen.
During the climactic battle with Gus' supercomputer, Superman again appears to be gasping for air whilst enveloped within an airtight sphere.
And you can see and hear Superman breathing at various other points in this movie, like right after the Kryptonite ray is shut off, or when he's digging people out of the rubble a bit later.
The strongest evidence is to be found within Supergirl (1984), which is set within the same continuity as the Christopher Reeve movies. Near the start of the film, a dragonfly created by Kara (using the Omegahedron) punches a hole in the barrier surrounding Argo City -- a Kryptonian city which remains intact on a fragment of the planet Krypton -- and air is clearly shown diffusing from the interior to the exterior, with sufficient force to suck the Omegahedron straight through the hole. Panic ensues until Zaltar reseals the hole.
The presence of a breathable atmosphere within the city is confirmed by subsequent dialogue:
ZOR-EL: You took the Omegahedron!
ZALTAR: That's not correct, I lost the Omegahedron!
KARA: Oh no, father, I did! I was--
ZOR-EL: No matter who. Without it, this city can't survive more than a few days!
ALURA: Our lights will grow dim, and the very air we breathe, so thin!