Answer:
Harper's Weekly
Edit - Most players did refer to Harper's in their responses. If the question had read "With which publication did Winslow Homer begin his career as a news artist sketching battle and camp scenes?" I would agree that Harper's would be the only correct response.
However, the question actually read "Where did Winslow Homer ...?" A number of players, including me, interpreted this to mean the geographic location where Homer went to observe the scenes that he included in his sketches, and gave our answers on the basis of that interpretation.
Since the wording of the question left it open to more than one interpretation, I am going to give credit for a correct response to everyone who answered.
Hoosier