The Washington-based Pew Research Center interviewed 10,000 people across Europe. Their answers showed that skepticism about the EU was not confined to Britain.
France viewed the bloc even more unfavorably than its neighbor across the English Channel, with just 38 percent of people having a positive opinion of the EU, compared with 69 percent in 2004. In Spain, approval of the EU has fallen from 80 percent to 47 percent.
Tim Oliver of the London School of Economics said a British vote to leave the EU could lead to demands for similar referendums across the continent.
“However, whether or not they succeed depends on, one, what happens to the UK," he said. "If the UK struggles, then that sets quite a powerful message to the rest of the European Union.
"And secondly, it depends on what happens in the Eurozone, it depends what happens in Schengen, it depends what happens with Russia. The European Union faces several crises at the moment, and Britain is one of them.”
Oliver added that a British divorce from the EU likely would be lengthy and bitter.
For the rest of the EU, “there has to be an economic cost" to the United Kingdom if it leaves the bloc, he said. "The UK can’t withdraw and not feel any punishment, any costs and so forth, and get a new relationship with the EU, or else that might encourage other member states to do the same thing.”
Pew researchers said much of the disapproval of the EU is related to its handling of the refugee crisis. More than 1 million migrants arrived in Greece in 2015; disapproval of Brussels’ migrant policies there runs at 94 percent.
A simultaneous so-called Brexit crisis could push Europe to the brink.
“They haven’t solved each crisis, they’ve just managed to cope and move on, and hope they’ll find a solution later on," Oliver said. "If several crises align at the same time, that puts the EU in uncharted territory, and that is quite dangerous.”
In a speech Wednesday, former British Foreign Secretary William Hague warned that a British exit could sow wider division.
“If we lift our eyes from our local concerns to the challenges coming our way within this continent and beyond it, then the risks of leaving are too great," Hague said at analyst group Chatham House. "Above all, the danger of a fractured, divided and weakened West is one we must do our utmost to avoid.”
And with disapproval of the EU growing across the continent, analysts say a British exit could have much wider consequences for Europe and beyond.