For 40 years, Mike DeWine rose steadily if blandly up the ladder of Ohio politics, finally landing his dream job as governor. He took office last year as a familiar figure in the state, not because of any indelible political identity, but because, at 72, he had been around forever.

But the coronavirus crisis has made Mr. DeWine something that decades in elected offices never did: a household name. A Republican, he took early and bold actions to lock down his state, even as the head of his party, President Trump, dismissed the threat of the pandemic.

Mr. DeWine’s decisiveness — closing schools before any governor in the country, postponing the state’s March 17 primary election to protect voters — sent his popularity soaring. The folksy governor, previously best known for an annual ice cream social at his rural home, became something of a cult figure on social media. Ohioans tuned into his five-day-a-week briefings to celebrate “Wine With DeWine,” a ritual whose motto is “It’s 2 o’clock somewhere.”

Now, Mr. DeWine is charting a way out of the shutdown, taking cautious steps while facing pressure from business leaders, conservative activists and some Republican lawmakers who vociferously question the economic costs of a state in quarantine.