“If it’s brown, it’s down” is a hunting saying that’s especially popular when deer hunting season opens. “If it’s brown, it’s down” has been cited in print since at least April 1990; there is no known authorship for the saying.
 
“If it flies, it dies” is sometimes given as an additional rhyme.
 
   
Google News Archive
24 April 1990, Milwaukee (WI) Journal, “Crawford County favors managing wild deer” by Ron Leys, pg. C5, col. 1:
While deer were managed else where by carefully calculated quotas of antlerless deer kills by management zone, in Crawford County the rule was “if it’s brown, it’s down.”
   
14 April 1992, Milwaukee (WI) Journal, “The Wisconsin outdoorsman proves himself to be smart, responsible” by Jay Reed:
“If it flies it dies, or if it’s brown it’s down,” is the way some of the thinking goes. It’s not true, of course. Never has been.
 
Google Books
Mackerel by Moonlight
By William F. Weld
New York, NY: Pocket Books
1998
Pg. 31:
“If it’s brown, it’s down” was the camp motto, and this applied to both deer and whiskey.
     
ArcheryTalk Forums 
Thread: Hunting Quotes and Sayings….
24ptnontypical
January 30th, 2003, 09:12 AM
My best buddy always says, “If it’s brown, it’s down”, but mine has always been, “Fill the freezer, then the wall”
 
Google Books
Deer Wars:
Science, Tradition, and the Battle Over Managing Whitetails in Pennsylvania
By Bob Frye
University Park, PA: Pennsylvania State University Press
2006
Pg. 176:
“If it’s brown, it’s down” was how some hunters expressed their fear that concurrent seasons would lead people to shoot at anything brown first and make sure it was a deer later.
 
Google Books
A Grouse Hunter’s Almanac:
The Other Kind of Hunting
By Mark Parman
Madison, WI: The University of Wisconsin Press
Pg. XII:
Deer hunters have a saying—“If it’s brown, it’s down”—and it’s small wonder that many among Wisconsin’s nonhunting populace fear walking in the woods during the nine-day gun deer hunt.