Although Captain America had a sizeable comicbook following prior to his real adventures, his two major (real world) escapades were both airborne; The infiltration of the Hydra Weapons Facility and his heroic suicide in the Hydra bomber, Valkyrie
Bearing in mind his close association with air power, it makes good sense for Steve (and the Howling Commandos, who evidently were also no strangers to air-drops) to be found in a museum devoted to...
...maintain[ing] the world's largest and most significant collection of
aviation and space artifacts, encompassing all aspects of human flight
Out of universe
Alas, the reasons for choosing the museum as a filming location are pretty mundane:
The Museum has a substantial tradition of working with the Hollywood studios
The Air and Space Museum was conveniently located to where the bulk of the film was being made.
Over 100 of the film's crew had already filmed extensively in the Smithsonian's Air and Space Museum in Virginia when making Transformers 3, meaning that they were knowledgeable about the museum's pre-conditions, safety regulations, contractual obligations, etc
For the record, it's worth noting that the museum's (real world) curator, Margaret Weitekamp has openly acknowledged the incongruity between their normal exhibits and the film's depiction of her museum:
Though the museum has curated pop-culture exhibits in the past,
Weitekamp says it has no plans to produce any tie-in programs with the
Avengers series. “Our focus is on spaceflight, and this is a very
earthbound film,” she says.