WASHINGTON—As Democratic presidential hopefuls promise not to boost their candidacies through spending by outside groups with cash from wealthy donors, sometimes given secretly, a new report says the party received most of the so-called “dark money” spent on political ads in the 2018 midterms.

That marked the first time since 2010 that liberal nonprofits outspent conservative ones, according to a report by Issue One, an organization that advocates reducing money in politics. Liberal groups accounted for 54% of the $150 million...