A National Base Ball Convention seems to be needed to settle the question as to the right of base ball clubs to pocket a couple of thousand dollars of gate money, and then turn the visitors out with only the satisfaction of knowing that if they pay their way in the next day they may see a game-if it does not happen to rain again. The thousands of people who took the chances of the weather yesterday to encourage the game of base ball, were entitled either to see a game, after having paid their money, or to have their money refunded. The former alternative would have been very easy for the managers, as they would have had nothing to do but to return the tickets they had taken and allow them to be used for to-day's game. Instead of this, however, they propose to keep the money, and to take in as much more to-day, if they can get it. Such a policy may be penny wise, but it is pound foolish, as well as dishonest. If the people who pay full price to see a game can be turned out of the grounds at the end of two or three innings, whenever it rains, they will take very good care not to subject themselves to the risk of rain, or of any other interruption, and the will largely avail themselves of the American privilege of staying away.-St. Louis Globe-Democrat, May 5, 1876
As always with things like this, we go to Peter Morris' A Game of Inches:
The origins of the rain check have been complicated by confusion over a couple of issues. The first is the distinction between the concept of a rain check and the means of distributing them. The second is that, as with Ladies Day..., the concept was experimented with many times before owners accepted that it made good business sense.
As a result, while it is often reported that rain checks were first used in baseball in the late 1880's, they are actually much older than that. The National League voted to end the practice on March 8, 1881, so obviously they were being used before then...The issue remained a contentious one, with the St. Louis Post-Dispatch reporting in 1883: "The crowd insisted that they should get back either money or rain-checks, but President [Chris] Von der Ahe refused to do either"...
It seems (if I'm reading Morris correctly) that the practice of issuing rain checks did become a common and accepted practice until the early part of the 20th century.