MANCHESTER, N.H.—Hillary Clinton said Wednesday she doesn’t support abolishing the death penalty but would like to see it used more judiciously, another point of contrast with the most liberal members of her party and with her nearest rival for the Democratic presidential nomination.

Her comments, her first on the death penalty during this campaign, came in response to a question at a forum here. It was one of three issues where Mrs. Clinton laid out differences with Sen. Bernie Sanders, who is challenging her for the nomination and mounting a particularly strong showing in New Hampshire. He opposes the death penalty.

Mrs. Clinton also reiterated her support for the Export-Import Bank, which the House just voted to reauthorize. She called the bank—whose charter expired amid opposition, mostly from Republicans—a critical tool for promoting U.S. businesses abroad. Mr. Sanders voted against reauthorization in the Senate, calling it “corporate welfare to multinational corporations.”

Mrs. Clinton also broke from many liberals, including Mr. Sanders, in discussing the future of Social Security. The left wants to expand benefits for most or all recipients. Mrs. Clinton said Wednesday she wants to increase benefits to the poorest, most vulnerable recipients—but she stopped well short of endorsing an across-the-board benefit boost. She also didn’t rule out other benefit cuts.

Her comments came on her first trip to New Hampshire since last week, when her campaign benefited from a string of events including Vice President Joe Biden’s decision not to run for president and her own well-received appearance before a House committee. Her poll numbers have been on the rise since then, though she still faces strong competition from Mr. Sanders, particularly in New Hampshire, where neither candidate has a clear polling advantage.