This answer was aimed at the original wording of the question, which was essentially that, if Euroskepticism seems to exceed the rational arguments against the EU, then what mechanisms are causing the EU's public relations to underperform?
Information warfare from Russia is the primary assailant of EU's stability and reputation
Russia's motivation for destabilizing the EU is to improve the global power balance between Russia and the EU, to allow Russia to steal land or assets without having to face sanctions, and to protect Putin's personal assets from foreign sanctions.
The BBC claims that Putin is using hybrid warfare (information warfare, cyber warfare, political warfare) to force the EU to drop sanctions against Russia.
This Reuters article also supports the narrative that Russia is waging a substantial information warfare campaign against the EU.
German intelligence officials, however, say there has been Russian support for euroskeptic, anti-immigrant parties in Germany and across the EU.
“We see disinformation attacks before every vote that is of interest for the Kremlin,” a second EU source said. “Very often the vote that follows ... turns out in favor of the Kremlin.”
This NBC article also details that Eurosceptic parties are enjoying the support of Putin's resources.
Under President Vladimir Putin, Russia is working to empower Europe’s far-right and Eurosceptic parties with offers of cooperation, loans, political cover and propaganda.
From the Russian side, empowering anti-European Union populists has the obvious benefit of helping erode the strength of European institutions that Moscow has long considered enemies.
Moscow’s entreaty to Austria comes only two years after a Russia-backed bank extended a nine million euro loan to Le Pen’s National Front. Facing two competitive elections this year, Le Pen has considered going back to a Kremlin-backed bank to fill her party’s $21 million shortfall.
Here is the wikipedia article on Russia–European Union relations. It details that...
- The Ukraine-Russia gas dispute damaged Russia's reputation and resulted in the EU upgrading Ukraine's pipelines. (Russia petroleum companies like Gazprom are also something Putin owns a significant personal stake in.)
- The EU imposed visa bans and asset freezes in response to Russia's military intervention in Ukraine.
- Russia views various NATO / EU applicants as being within its sphere of influence, and therefore feels that NATO is creeping closer to Russia's borders.
- "Moscow increased its efforts to expand its political influence using a wide range of methods, including funding of political movements in Europe, increased spending on propaganda in European languages, operating a range of media broadcasting in EU languages and web brigades, with some observers suspecting the Kremlin of trying to weaken the EU and its response to the Ukrainian crisis."
- "Russia has formed close ties with Eurosceptic and populist parties belonging to both ends of the political spectrum. By the end of 2014, a number of European far-right and far-left parties were receiving different forms of financial or organisational support from Russia in an attempt to build a common anti-European and pro-Russian front in the European Union."
- "Konstantin Rykov and Timur Prokopenko, both closely tied to United Russia and Russian Federation’s Presidential Administration, were the key figures in funneling money to these parties. Agence France-Presse stated that "From the far right to the radical left, populist parties across Europe are being courted by Russia's Vladimir Putin who aims to turn them into allies in his anti-EU campaign" and that 'A majority of European populist parties have sided with Russia over Ukraine.'"
- "A significant part of the funding of anti-EU and extremist parties passes through St Basil the Great fund operated by Konstantin Malofeev."
- "In June 2015, the European Parliament stated that Russia was "supporting and financing radical and extremist parties in the EU" and called for monitoring of such activities. France's National Front, UKIP, and Jobbik voted against the resolution. These and other extreme right organisations are part of Russia-sponsored World National-Conservative Movement."
- "an journalist Olga Dragilyeva stated that "Russian-language media controlled by the Russian government and NGOs connected with Russia have been cultivating dissatisfaction among the Russian-speaking part of the population" in Latvia."
- "In 2015, the U.K. media said that the Russian leadership under Putin saw the fracturing of the political unity within the EU and especially the political unity between the EU and the U.S. as among its main strategic goals, one of the means in achieving this goal being rendering support to Europe's far-right and hard Eurosceptic political parties."
- Quite a lot of other claims about Russia's efforts to destabilize the European Union, far too long to list here. If this abridged summary is still not satisfactory to you, please visit the linked article.
Movements to Withdraw from the European Union
Russia tried to weaken the EU by interfering in the Brexit referendum, which would both shrink the EU's political power, as well as economically and socially destabilize both the EU and Britain.
Evidence for Motive
Below is evidence that Russia's motives to conduct an information warfare campaign include the financial interest of its leaders and oligarchs as well as interest in using destabilization and influence efforts as retribution for sanctions. It also lists kinds of sanctions that Russia is facing that it would wish to make go away.
Russia faced international sanctions for its annexation of Crimea from Ukraine.
Russia's recent geopolitical interference also includes military intervention in the Syrian Civil War.
Russia's human rights violations also include anti-gay genocide in Chechnya, which also led to sanctions against five Russian leaders and oligarchs, as ordered by the U.S. Treasury Department.
Another example of a possible sanction against Putin is the Magnitsky Act, which sought to sanction Putin for the murder of an investigator who discovered a $230 million theft perpetrated by Putin.
A likely motive for Putin's global meddling could be in order to protect his financial assets, which have often been estimated to be valued at around $40-90 billion, may have been largely accumulated through theft from the Russian people, and may be at risk of asset freezes from foreign authorities like the United States (in this source's case, the U.S. under Barack Obama's leadership).
On another note related to Putin committing foreign meddling to protect his own interests, Putin blames Hillary Clinton (while Secretary of State) for inciting unrest in Russia, nearly costing Putin defeat in an election. It is suspected that this was a primary motivation for Russia's interference to hand Clinton an election defeat in 2016 (in addition to installing an ally who could help steer NATO for Russia's benefit), and of course, evidence supporting Russia's interference in the election is thoroughly substantial. Trump has also championed Euroskepticism since assuming office.
Another form of sanction that Russia is facing is ironically punishment for the U.S. election interference itself.