FRANKFORT, Ky. — In the windowless nerve center that resembles a campaign war room, Gov. Steven L. Beshear studied projections on a wall showing that 600 people were logged on to the state’s health insurance exchange.

Some 34,000 had begun applications, and more than 11,000 had signed up for plans, making Kentucky one of the most successful state-run insurance marketplaces under the new federal health care law.

“You are all doing a fantastic job,” Mr. Beshear told two dozen bleary-eyed workers.

In a state where dislike of President Obama runs strong and deep, Mr. Beshear, a Democrat, has positioned himself as a champion of the Affordable Care Act, out ahead of public opinion. It has endeared him to the White House at a time when news of the problem-plagued federal exchange, HealthCare.gov, has been embarrassing and damaging.

“My message to Kentuckians is simply this,” Mr. Beshear said in his office in the State Capitol. “You don’t have to like the president; you don’t have to like me. Because this isn’t about him, and it’s not about me. It’s about you, your family and your children. So do yourself a favor. Find what you can get for yourself. You’re going to like what you find.”