Democratic voters are showing a higher level of interest in the 2018 midterm elections than are Republicans, with three-quarters of Democrats saying their vote for Congress is intended as a message of opposition to President Donald Trump, a new Wall Street Journal/NBC News poll finds.

While Democratic interest in the election rose from March, the spike didn’t translate into gains for the party in a broader measure of the political climate: Asked which party should control the next Congress, voters favored Democrats over the...