When the Democratic Party … sees fit in its wisdom to nominate a yaller dog … I will support … — but

The term is now more generally applied to refer to any Democrat who will vote a straight party ticket under any circumstances.
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The phrase "Yellow Dog Democrat" is thought to have achieved popularity during the 1928 presidential race between Democratic candidate Al Smith and Republican candidate Herbert Hoover, when Senator J. Thomas Heflin (D-Alabama) crossed party lines and formally supported Hoover. Many Southern voters disliked several items on Smith's platform but still voted for him.
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The first known usage to date of “yaller dog” in relation to Democrats occurred in the 1900 Kentucky gubernatorial contest involving Kentucky Governor William Goebel. Theodore Hallam was criticized at a Democratic Party meeting for first supporting Goebel, then campaigning against him. The critic pointed out that Hallam earlier had said "if the Democrats of Kentucky, in convention assembled, nominated a yaller dog for governor you would vote for him" and asked "why do you now repudiate the nominee of that convention, the Honorable William Goebel?" Hallam responded:

"I admit," he stated blandly, "that I said then what I now repeat, namely, that when the Democratic Party of Kentucky, in convention assembled, sees fit in its wisdom to nominate a yaller dog for the governorship of this great state, I will support him — but lower than that ye shall not drag me!"

https://en.wikipedia.org/wiki/Yellow_dog_Democrat