Origin of the Phrase
According to Christine Ammer, The Facts on File Dictionary of Clichés, second edition (2006), the phrase has been around in the United States since approximately 1930:
put one's money where one's mouth is Back up your stated position with action. This term, according to Eric Partridge's informants, was current in the United States from at least 1930 and caught on in Great Britain and other English-speaking countries shortly after Wold War II. In 1975 the British government used it as an advertising slogan to persuade people to invest their savings in the National Savings Bank Accounts Department.
As recently as Eric Partridge, A Dictionary of Slang and Unconventional English, eighth edition (1984), however, Partridge's informants had not pushed the coinage date earlier than 1945:
put your money where your mouth is! Back your words with cash (cf. the orig., U.S. sense of put up or shut up!); since ca. 1945.
Early Occurrences Found in a Google Books Search
A Google Books search turns up three pre-1930 matches for the phrase. From Howard Washingtom Odum, Rainbow Round My Shoulder: The Blue Trail of Black Ulysses (1928):
“How much you got to put in the game?”
“Hell, that’s not enough.”
“You are a dam’ fool, how much you got?”
“I have ten dollars.
”Well, we can have a good game. Get the cards then and let’s meet the dam’ gang.”
“Deal the jack of di’mon’s, that’s my card.”
“Bet your money, go to hell.”
"Put your money where your mouth is."
"It is down, turn them dam' cards you have fell."
From Trans-communicator, volume 45 (1928) [combined snippets]:
The non is still with us. You can see him strutting his stuff at some of our positions or hear him on the wire making various inquiries as to when he will receive a vacation or increase in pay. Such persons should be told to put their money where their mouths are and they will get results. Just a little more work by us all will go a long ways towards removing these pests, and each of you have been furnished with a list showing these nons. And their locations are well known, so make another effort to try to Induce them to join as ...
And from Trans-communicator, volume 46 (1929) [snippet]:
There are several proposals up for general benefit, but to put them over our support is needed and one of our man ways of supporting our committee is to "put our money where our mouth is—and "pay up." Lafayette District — Bro. Hetherwick
But even older are “put their money where their faith is” (in Methodist Episcopal Church Year Book, 1881), “put your money where your interests are” (in The Railroad Telegrapher, page 1291, September 1905) and “put your money where your heart is” (in Publicity and Progress, 1915, and—as “put his money where his heart is”—in The Harvester World, April 1919).
Perhaps one or more of these other phrases influenced the emergence of “put your money where your mouth is.” Or perhaps the notion of putting your money somewhere is so natural and commonplace in modern English that it was only a matter of time before the mouth came up as a suggested destination.
For a discussion of the older, possibly related phrase "put up or shut up," see What is the origin and sense of the phrase “put up or shut up”?