Cristopher Clark, in his book The Sleepwalkers about the run-up to WW1, devotes a chapter each to the pre-war situation in the countries whose conflict originally sparked the war (Serbia and Austria-Hungary). He argues that late-stage Austria-Hungary was far from "on the verge of collapse", as it was long portrayed, and discusses some of the reasons why it got a bad rep.

First for some background: the Austrians were decisively kicked out of Germany when Prussia inflicted a crushing military defeat on them at Königgrätz (1866). With the loss of its influence among the German states, Vienna was forced to shift its attention to the south and east instead. The Hungarians seized on this opportunity to demand more political power within the empire, leading to the Compromise (Ausgleich) of 1867, in which the Austrian Empire was transformed into "Austria-Hungary", with the two countries enjoying equal status under a common monarch.

That being said, the empire wasn't just populated by Austrians and Hungarians, but also by Czechs, Croats, Slovakians... All in all there were eleven officially acknowledged nationalities, which, needless to say, often led to tensions. The Hungarian elite in particular (who had themselves long gotten the short end of the stick from the Austrians and the Turks) pushed an aggressive nationalist policy that opposed or ignored the demands of other ethnic groups within "their" half of the empire.

Clark acknowledges that the bizarre political system set up by the Compromise wasn't the smoothest-run machine, but he calls this 'a chronic ailment, rather than a terminal disease,' adding that:

Balancing the complex array of forces that resulted [from national and ideological conflicts within the empire] to sustain a working majority was a complex task requiring tact, flexibility and strategic imagination, but the careers of the last three Austrian prime ministers before 1914, Beck, Bienerth and Stürgkh, showed – despite intermittent breakdowns in the system – that it could be done.

Similarly, he sees the problems caused by competing nationalities as surmountable:

...case-by-case adjustments to specific demands suggested that the system might eventually produce a comprehensive mesh of guarantees for nationality rights within an agreed framework. And there were signs that the administration was getting better at responding to the material demands of the regions.

Other positive aspects he points to are Austria-Hungary's fast industrialisation in the period 1887-1913, its well-functioning bureaucracy and infrastructure, and last but not least, its role as a guarantor of peace and security among peoples who might otherwise be at each other's throats.

Where, then, did the popular image of Austria-Hungary as a hilariously dysfunctional state about to collapse under its own weight come from – in other words, what were the "propaganda reasons" you mention?

The Entente obviously had an interest in the image of Austria-Hungary as coming apart at the seams – in Clark's words, 'hostile contemporaries' pushed this view in order to 'suggest that the empire's efforts to defend its integrity during the last years before the outbreak of war were in some sense illegitimate.'

Moreover, Clark argues that Austria-Hungary's ethnic makeup, aside from the real problems it caused, made it an easy target for contemporary mockery – to the prevailing nationalist mindset of the day, it seemed almost natural that this weird patchwork empire must be doomed to failure.

And last but not least, hindsight is 20/20. Austria-Hungary collapsed in WW1; so in hindsight it was all too easy to see reasons why this collapse was inevitable everywhere. Clark cites one particularly poignant example: the British journalist Henry Wickham Steed, who in 1913 had written that Austria-Hungary's 'internal crises are often crises of growth rather than crises of decay,' and that he did not see 'any sufficient reason' why Austria-Hungary 'should not retain its rightful place in the European Community.' Years later, the same Steed would write that he left Austria-Hungary in 1913 'with the feeling that I was escaping from a doomed edifice.'

TL;DR: Late-stage Austria-Hungary wasn't without its problems, but they seemed surmountable and few people at the time saw signs of imminent collapse. The image of Austria-Hungary as doomed to failure mostly came about because A) it made the Entente look better, B) the empire's multi-ethnic makeup ran against the Zeitgeist and C) big historical events have a tendency to seem inevitable once they've actually happened.

My source is mostly The Sleepwalkers itself, but also Die Ungarn by Paul Lendvai for some background from the Hungarian perspective.

I will propose some alternate ideas to the esteemed midnightrambulador, and will draw on the impressions of Austria I gained from A.J.P Taylor's 'The Habsburg Monarchy'

First, Austria was heavily dependant on the Emperor to actaully get things done, a problem when you have the unimaginative, and by 1914, decrepit, Franz Josef in charge

Second, the compromise had empowered the Magyar nobility, and they were now focused on turning the other nationalities of Hungary into Magyars, while utterly refusing any possibility of reforms that might harm their own power. A side-effect of this was chronic underfunding of the Common Army, because it did not directly serve Hungarian interests. This would seriously come back to bite them. Magyar intransigence also meant that there could be no reforming into a tripartite structure, advancing a Slavic group, because it would come at the expense of Magyar power either by empowering the Czechs as counter-balance, or breaking off Croatia from Hungary. While few things in history is inevitable, and pre-war the majority of Czechs were still loyal, there were tensions caused by rising Czech nationalism and the language disputes in Bohemia, as well as rising Yugoslav sentiment (and Serbian agitation) in the Balkans, especially after the annexation of Bosnia in 1908.

Third was the issue of the parliament in the Austrian half of the Empire. The franchise had expanded to remove property requirements and become universal by 1906. What this meant though, was that the formerly middle-class German speaking electorate became fractured along ethnic lines, with each nationality electing nationalist representatives, who each sought improved status for their ethnic group, while the German-speaking vote began to produce pan-Germanists who wanted to dissolve the empire and join Germany. On top, the only recognised language in the Council was German, which led to filibusters by non-Germans in their own languages, which led to shouting matches and chaos. Notably, the council was dissolved before WWI even broke out, Taylor describing how it ended with lengthy speeches by Ruthenian delegates and the throwing of inkpots. In Hungary, by contrast, only 6% of the population had the vote, to ensure power remained in the hands of the Magyar nobility, and away from the working classes and the non-Magyar ethnic groups

Fourth, Austria's international position rendered it weak. Italy had obvious claims on its territory, while Russia opposed it for primacy in the Balkans. The Russian threat was especially severe and likely to provoke war, as Bismarck observed, 'over some damn fool thing in the Balkans'. Austria was forced into dependance on Germany, to whom it was an obviously junior partner. The ineffectiveness of the KuK army eventually led to Germany taking over most of the management of the Austrian state.

Essentially, Austria's main problem was its composition. In an age of nationalism, Austria was always going to be vulnerable to unrest, especially once universal voting and mass literacy entered the scene. Without a war, Austria could have happily chugged along, but any was likely to put internal strain that broke it apart. Not only that, but the calcification of the European diplomatic system into rigid alliances removed the traditional Habsburg tool of careful diplomacy. Austria could no longer change partners and walk a delicate line, it had to cling to Germany due to its situation in the East with Russia and Serbia, which itself led to an inevitable clash with the Western powers with which it actually had no quarrel, but that Germany's annexation of Alsace-Lorraine all but guaranteed.

Tl:dr Austria-Hungary wasn't about to collapse by itself, but any sort of major strain was about to bring tensions to boiling point, which is what happened