New York City (or Thereabouts)
His Girl Friday never tells you precisely what city you're in… which is probably a smart move on their part, because the film portrays the city government as insanely incompetent and corrupt.
But here's what we do know: it's somewhere in New York State, since Albany is the state capital. And how many other huge, famous, media-centric cities are there in New York state besides The Big Apple? Not a one. (Syracuse doesn't count, guys.)
It's important that the setting be New York—or some similar big urban center—because you need a big urban center to contain Walter and his fast-talking, dynamic personality. He's just too big for a small town.
And after all, where could you find a plane to write in the sky begging your wife not to divorce you—except in New York? Where else would a newspaperman have two real life criminals on the payroll—except in Gotham? Where else would you pick up the phone and declare—
Now listen, you ten-cent glamour girl. You can't keep Butch away from his duty!... What's that?... You say that again, I'll come over there and kick you in the teeth!...
—except in The City That Never Sleeps And Is Rude Because It Is Sleep-Deprived?
New York is fast, gigantic, chaotic, and profane. If you insult people three times before you even shake their hand, you're in a New York state of mind.
In the film, New York is directly contrasted with Albany. Bruce is from Albany, and Hildy wants to get there for peace, quiet, and domesticity.
BRUCE: Mighty nice little town, Albany. They've got the state capitol there, you know.
Albany is little: pretty much all they've got going for them is the state capitol. (According to this movie. No offense, Albany.)
Albany's hardly enough to contain Hildy or Walter—though they will visit to report on a strike, maybe, just to bring the news back to New York.