Democratic presidential candidates are presenting policy ideas that are broadly popular with Americans, including tuition-free state colleges, but other proposals—such as Medicare for All—could complicate the party’s prospects next year, the latest Wall Street Journal/NBC News poll shows.

Two-thirds of registered voters support letting anyone buy into Medicare, similar to an idea that former Vice President Joe Biden and some other Democratic candidates have proposed. Two-thirds say that young adults brought to the U.S. illegally should be allowed to stay, an idea broadly supported by the party’s presidential field. Nearly 60% of registered voters support making tuition free at state colleges and universities.

But several other ideas backed by majorities of Democratic voters and some of the party’s 2020 candidates draw significant opposition from the electorate overall, the new poll finds.

Some 56% of registered voters oppose a Medicare for All plan that would replace private insurance, as Vermont Sen. Bernie Sanders and some others have proposed, while 57% oppose the idea of immediately canceling student-loan debt for all borrowers. Mr. Sanders also has proposed the latter, while Massachusetts Sen. Elizabeth Warren backs it with limits.

Of 12 policy ideas tested in the poll, providing government-sponsored health care to undocumented immigrants was the least popular among the broader electorate, with 62% rejecting it. In a June Democratic debate, all 10 candidates on the stage, including Messrs. Biden and Sanders, raised their hands when asked who backed the idea. In the new survey, it was supported by 64% of Democratic primary voters but only 36% of voters overall.