Democratic strategists once sidestepped the issue, seeing any proposal that might raise energy prices as politically risky. But they are now pushing it to the forefront.

At the convention, organizers played a short film by James Cameron, the director of blockbusters like “Titanic,” on the dangers of climate change. Another convention video montage put a spotlight on Mrs. Clinton’s role at the 2009 climate change summit meeting in Copenhagen. As part of a wide-ranging climate plan, she has set ambitious goals for producing energy from renewable sources, including by installing a half-billion solar panels by 2020.

Democratic Senate candidates in swing states like Pennsylvania and Florida are also embracing the issue. They have been emboldened by polls showing that a growing majority of Americans accept the science of climate change and would support candidates of either party who vowed to address the issue.

A Gallup poll in March found that 65 percent of Americans believed that climate change was caused by human activity, an increase of 10 points from a year earlier. The poll found that 38 percent of Republicans believed the same thing, an increase of four points from a year earlier. The poll also found that 76 percent of Americans ages 18 to 29 accepted that human activity is behind climate change.

Some Republican strategists say they are concerned that Mr. Trump’s views on the issue could push younger voters away from the party for the long term, much as they fear that his immigration policies and remarks about women could alienate Hispanics and female voters.

“It’s important for Republican candidates to talk about the issue intelligently and not be dismissive of climate change,” said Whit Ayres, a Republican pollster. He worked for the presidential campaign of Senator Marco Rubio of Florida and now works for an advocacy group backing Mr. Rubio’s Senate re-election campaign.

“The way you talk about climate change sends a signal to millennials about how sensitive you are to the environment,” Mr. Ayres said. “Millennials recently passed baby boomers to become the largest generation, so any party that hopes to own the future politically needs to be attractive to millennials.”