One of the themes in "The Death of the Ball Turret Gunner" is the dehumanizing aspect of war, and we are not meant to understand enough about the gunner to give him an identity separate from his identity as part of the war machine.
The first two lines, for example,
From mother's sleep I fell into the State/And I hunched in its belly till my wet fur froze. . . .
truncate the gunner's life to only two critical events, birth, and life as part of the State. Jarrell undoubtedly capitalized "State" to reinforce the message that the ball-turret gunner's only useful life is as a part of the war machine of the United States, and, in this case, the "State" is represented by the ball turret in the belly of a bomber.
Not only does the ball-turret gunner have no identity as an individual, he's described as being in the State's belly "till my wet fur froze." His wet fur, of course, is the fur lining of his flight suit, wet from the sweat of intense fear. But the image of "wet fur" is meant to further remove the gunner from humankind--the wet fur is, in part, an animalistic metaphor that helps Jarrell create the image of the gunner as part of a machine rather than a distinctly human individual.
In sum, then, the "State" represents the anonymous war machine to which the gunner is merely an appendage, suspended in the belly of a bomber and completely detached from the normal world of men.