MORRISTOWN, N.J.—A ficus tree sits in a corner of the Morris County Democratic headquarters as a symbol of a top party priority in the 2018 midterm elections—unseating one of the U.S. House’s most powerful Republicans, Appropriations Committee Chairman Rodney Frelinghuysen.

The plant is a homage to liberal filmmaker Michael Moore, who once ran a write-in campaign for a ficus against the New Jersey congressman to highlight the absence of competition against entrenched incumbents. Mr. Frelinghuysen’s was a safe district, in which Democrats had never put up a well-funded opponent.

Now Mr. Frelinghuysen, 71, is facing the first serious challenge of his 23-year congressional career, a race that stands as a barometer of the GOP’s 2018 headwinds. Even once-secure Republicans at the pinnacle of congressional power appear at risk.

Mr. Frelinghuysen has drawn a field of opponents who say he has grown remote and out of step with his district. His leading Democratic opponent, who raised more money than he in one quarter last fall, is Mikie Sherrill, a former Navy helicopter pilot, former prosecutor, political novice and mother of four.

Mr. Frelinghuysen has taken heat at home for voting to repeal the Affordable Care Act and from Republicans in Washington for voting against their tax-cut bill. He has faced weekly protests at his district office, organized by a liberal group called NJ 11th for Change to draw attention to his decision not to hold open town-hall meetings.