In responding to President Trump’s address to Congress on Tuesday, Democrats had to navigate between the expectations of their angry base in America’s cities and the need to appeal to a broader array of voters in parts of the country where the president is far more popular.

The party handed that task to former Gov. Steven L. Beshear of Kentucky, an emblem of the sort of largely rural state that Democrats lost in last year’s presidential election. Delivering the party’s official response, Mr. Beshear, dressed in khakis and a blue shirt, sat in a Lexington diner and offered down-home references to Friday night football, Sunday morning worship and life as a preacher’s kid.

Mr. Beshear noted that the Americans who had gained health insurance under the Affordable Care Act, a law he championed in his state, were the sort of “friends and neighbors” he surrounded himself with in the diner.

Before the law was passed, “they woke up every morning and went to work, just hoping and praying they wouldn’t get sick,” he said. “Because they knew they were just one bad diagnosis away from bankruptcy.”