The thing is, "Rum, Romanism and Rebellion!" was not some random and peripheral remark. As David T's post suggests, times were moving on from the Civil War era and new issues were rising, giving old factions that had been suppressed (not, in my humble opinion, without good reasons in some cases) new allies. The alignments were shifting. The Republicans themselves were factionally splitting between former alliances, with agrarians with strong notions of their fundamental rectitude (and thus ownership of the lofty principles of Lincoln's party and his aura) polarizing ever more bitterly against the Gilded Age "Robber Barons/Captains of Industry" another branch of the party figured were Lincoln's important legacy. Cleveland as a candidate projected a powerful (and I gather, well-founded) image of propriety and honesty. After all we are talking about a man who confessed, when challenged, to having had a relationship with a woman he was never married to and taking responsibility for supporting her out of wedlock child. No one knows if he was in fact this child's father--he acknowledged he might have been, but could not tell because the mother shared her affections with others in his then circle of friends--he being the only bachelor among them, he took the responsibility to spare them ongoing scandal. Republican smear campaign (albeit founded on truth)--"Ma, Ma, where's my Pa?" Jubilant Democratic rejoinder after the election--"Gone to the White House, Ha Ha Ha!"
In connection with another thread I took a look at Cleveland's bio some time recently, and was impressed with the quality of his integrity. That said were I ISOTed to the era I'd want to back a more radical figure; he was quite conservative in many ways. But while some decades before I'd favor the Republicans strongly, by this time the rot in the established, insider figures in that party (never too clean-handed to be sure, but the dirt was more creative earlier on) was pretty deep; especially given the drift of OTL history an uptimer like me is gifted with knowing, I'd want to push for a really radical reformer. Certainly the post-Reconstruction Democratic Party would be a very dubious star to hitch my righteous wagon to.
But the thing that strikes me about "Rum, Romanism and Rebellion" is its mean-spiritedness, its essentialism, its very un-Lincolnesque smugness that there are saved elected (good, well-off Protestant Republicans coasting on the glory of the Union in the past Civil War--without following through on seeing to the welfare of the former slaves any longer) versus a mass of the presumably damned. The complacency and tribal mentality shows I think that whether Blaine might have won in 1884, sooner or later there would be a reckoning of some kind.
Note that the Republican victory that divided Cleveland's two terms into non-consecutive ones was very blatantly a stolen one. Just three times in US history since the modern party system evolved (basically with Andrew Jackson) has the popular vote been contradicted by the electoral vote. One of these, the Election of 1876, is of course the widely known textbook case of a stolen election--though in this case, I personally think the Republicans were justified, since the newly rising Solid South depended on mass voter suppression of African American votes--had this not happened, Hayes would clearly have won a larger popular vote, and won electoral votes from Southern states that as things stood counted their African population for purposes of getting representatives in Congress and thus electoral votes, but did not allow that segment (and a fair number of whites either) to actually vote. Therefore stealing Oregon's vote back seems not too outrageous to me. (The outrageous part is Hayes agreeing to placate the controversy by ending Reconstruction--but after all it was bound to be abandoned someday).
The third case was of course the election of 2000. I won't expound on that--wrong forum after all!
And the forgotten second case? The "victory" of Harrison in 1888, known at the time and to history to have depended on corrupt manipulations in favor of the Republicans in certain states. A number of bad things came out of the Harrison administration, and Cleveland handily swept him aside in 1892.
I suspect a Blaine administration would have been as infamous as the Lesser Harrison's deserves to be. The other thread that interested me was a South American focused one that (rather improbably IMHO though it was an interesting thread to follow, and presumably may still be if it revives) involved Chile forming a grand alliance of South American powers against the Yankees, confronting them in Panama. Because Cleveland was President at the time it drew my attention to his administration, and his predecessors during the preliminary Pacific War, in which Chile seized control of the former coastlands of Bolivia as well as some Peruvian territory. As it happened, the USA under the prior Republican administration favored the losing side in that war. Blaine was Secretary of State early on and spoke (perhaps hyperbolically) of annexing Peru--he was replaced, I suppose for other reasons, by a more level-headed figure. But learning about this did nothing to improve the rather negative picture I had of Blaine before. Cleveland on the other hand opposed imperialism, though he did cave in against overwhelming Congressional opposition on such matters as the status of Hawaii's Queen Lilioukulani. At the same time Cleveland also acted to revamp the rather uninspiring weakness the US Navy had fallen into--again, a Navy that was something to be reckoned with in Civil War days had, by virtue of corruption, underfunding and coasting on Civil War glories by simply retaining the same once-cutting edge ships as they were surpassed by the advancing state of the art fallen into third-rate status. It was widely believed, probably with foundation, that a single battleship Chile had purchased from Britain would be able to sink every ship in the then USN with impunity. Cleveland acted to rectify that, though the results would take years to bear fruit.
(This is why it seemed rather unfair to me for the Chileans to choose to challenge El Norte at the particular moment they did. Sending that battleship to Panama, then part of Colombia, was apparently OTL. It did happen on Cleveland's watch--though the US Admiral present withdrew instead of creating a POD by ordering a mindless attack on it against his senior staff officers' collective advice).
So yeah, Blaine might win. But the "gaffe" that might have been crucial in costing him this election was the tip of an iceberg that dominated the political landscape. Thus the infamous phrase might not have been crucial to the outcome after all. And if chaos in this closely balanced election had put Blaine over the top, the matters would come to a head sooner or later, I believe.