The fate of Reconstruction being at stake was certainly understood by anyone listening to Presidential campaigns and to general media.

The thing is, the reason for that was because both parties effectively backed the end of Reconstruction. The Republican Platform of 1876, the platform of the "party of Lincoln" so to speak, said this:

The permanent pacification of the Southern section of the Union and the complete protection of all its citizens in the free enjoyment of all their rights, are duties to which the Republican party is sacredly pledged

However, this was not the ultimate place where the party's goals stopped. As Eric Foner put it in Reconstruction: America's Unfinished Revolution, the "abandonment of Reconstruction was as much a cause of the crisis of 1876-77 as a consequence". He writes that Republicans were no longer willing to intervene in defense of black rights, and that as a result, Tilden (the Democratic candidate) would never have come close to carrying the South. In his gubernatorial campaign in 1867, Hayes (and others in their own elections) had put out strong support for black suffrage for example, and the Republican party had been thus committed. By 1876, support for the "ultra measures" to restrain the South had waned among Northerners, and Hayes took note. In his acceptance letter for the nomination, he himself noted that he would labor to allow the South "to obtain for themselves the blessings of honest and capable local government", which he clearly understood to mean the end of Reconstruction.

Voters by and large knew this, and accepted it. It was based on what Hayes and others thought they would want in the first place. It was for that reason, of course, that Frederick Douglass asked Republicans at their convention, point blank, "Do you mean to make good to us the promises in your constitution?". He knew, and others knew, that Republican commitment to Reconstruction had already begun to wane.

Democrats were, of course, already well committed to ending Reconstruction and to the restoration of more Southern autonomy in local politics. The removal of troops would be a large boon to those seeking that end. It should not come as a surprise that the Democratic Platform emphasized the need "to be saved from a corrupt centralism". The feeling of abandonment pervaded Southern freedmen, leading to a sense of pessimism made worse by the interference and violence that many armed bands of Democrat-supporters inflicted on Republicans in the South. The level of tension rose significantly, as the election's battle lines led to black Democrats being referred to as traitors to other African-Americans. Even so, there was little hope that this type of intimidation could outdo the intimidation caused by Democrat electoral interference. After all, as historian William Gillette wrote in Retreat from Reconstruction:

...the reaction of most northern Republicans [to the removal of troops from the South] ranged from enthusiastic relief that the issue of the use of troops in the south would no longer intrude into every campaign, to fatalistic acceptance of the necessity of withdrawal

People knew the fate of Reconstruction was likely at stake. Republicans did, Northerners did, and many freedmen did as well. They still generally bore an affinity for the Republican party, but it was clear that the Republican party's affinity for them was only going to extend them so far, and it was also clear that the "nonentity" that Hayes was meant less forceful support for Reconstruction, even if he won. So why did Northerners stop caring? Well, at least part of the reason involved the economic crisis starting in 1873, which led Northerners to conclude that Southerners should be left to their own devices, while the rest of the country focused on issues they considered more salient, like the economic crisis. This was coupled, some scholars have argued, with the rise of press accounts depicting white Southerners as "oppressed" by the African American population. The Democratic press's "masterful job" engineering this narrative, combined with economic crisis, led to a loss of enthusiasm for Reconstruction, according to Thomas J. Culbertson, of the Hayes Center (who does, in some part, claim Hayes has been cast too strongly as a villain, saying Hayes did not truly "end" Reconstruction). While Hayes's personal responsibility is debatable, what is fairly clear is that Reconstruction's fate was likely doomed eventually, without some strong show of leadership that simply never came. And the election was clearly less concerned with Reconstruction as a factor for debate or dispute.

Another fascinating place to look is at media reports from the time. Democrat papers were, right after the election, already claiming that African Americans were allowed to double-vote. This would form part of the narrative (that had already begun to grow) that was used to justify Jim Crow policies and the like following the end of Reconstruction, which depressed black turnout to 2% or so by the early 1900s. The National Republican ran some articles noting that "Unfortunately, Republicans themselves have in some notable instances assisted in transferring power to the hands of Democrats", regrettable (they argue) because it led to the electoral crisis of 1877 that also was the culmination of efforts to "subvert the freedom of the ballot in the South".