There was a Protestant minister who introduced him at a New York City Republican fundraiser who, on that occasion, described the Democrat Party as the party of "Rum, Romanism, and Rebellion." That was of course a none too subtle expression of anti-Catholic bigotry and specifically anti-Irish bigotry. It took the GOP nearly a century to put that behind them along with the Massachusetts GOP equivalent that greeted Irish immigrants on the docks at Southie (South Boston): "Jobs available: No Irish need apply."
I don't know anything about money corruption of the conventional sort (see Robamney for an example) that may have touched or defined Blaine. I'll leave that to others.
Grover Cleveland was a champion of small government. He may have been the last POTUS to actually use the Tenth Amendment regularly in vetoing any Congressional enactment (and there were many) that he viewed as not being characterized as among the specific powers granted to the federal government by the Constitution itself. He was a hard money man who was the opposite of William Jennings Bryan in that respect. You may say that he was a champion of "the flag, the Bible and no damn taxes." To this day, Grover Cleveland is responsible for the fiscal conservatism of French Canadians in the New England states for reasons that are not clear to me.
When Benjamin Harrison defeated Cleveland in 1888 after Cleveland's first term, Senator Harrison famously went to the office of his campaign manager (Matt Quay) on the morning after the election and said: "Matt, thank God, we have won." Quay replied: "Harrison, God had nothing to do with it. We bought every vote you got." Cleveland never bought votes, not even with taxpayers' money. Four years later, his honesty and frugality were missed and Cleveland was again elected to the White House for his second and final term.
As to the corruption allegations, he had been tied to the Credit Mobilier scandal (though never definitively proven)and the worst one was essentially a bribe made to him from Union Pacific over the “purchase” of some worthless railroad bonds. Though it apparently wasn’t definitively proved, there was correspondence (the Mulligan Letters) in the matter that was enough to raise legitimate suspicion.
The infamous chant about Blaine stemmed from those letters, in which he cited in one: “burn this letter !” The entire chant went, “Blaine, Blaine, James G. Blaine: Continental Liar from the State of Maine, Burn this letter !”
Of course, too, Rev. Samuel Burchard’s ‘Romanism’ speech didn’t help matters and incensed Irish Catholic voters (the irony being Blaine’s mother was Catholic and he was viewed as anti-British, two points that Blaine expected to help him amongst Irish Catholics !). It’s quite possible that had the latter speech not been given, Blaine would’ve pulled out a narrow victory over Cleveland (and had he not declined a run in 1888, probably would’ve won as Benjamin Harrison did).