By Collin Eaton

Businesses from factories and offices to salons and bars, once hopeful about a smooth reopening this summer, are now grappling with whether to close, stay open or find some in-between as the number of cases of Covid-19 increases in dozens of states.

Apple Inc., which said Friday it would close nearly a dozen stores in four states, said Wednesday it would shut seven more in the Houston area, where cases have doubled so far this month. Restaurants around the country that recently reopened have closed again for anywhere from three days of deep cleaning to two full weeks so staff could self-quarantine after outbreaks.

In California, where new cases reached new highs Tuesday, workers for Walt Disney Co. pressed the company to delay the planned July reopening for its theme parks in California and Florida, saying the company is forcing them into unsafe situations. Disney responded late Wednesday, saying that it would postpone the scheduled reopening of its park in Anaheim, Calif., which had been set for July 17.

Disney has said the safety of workers and guests is at the forefront of planning. Disney is also considering postponing the July 24 release of "Mulan," the first major studio release on the schedule, as the virus complicates plans to reopen theaters.

Mounting signs that the spreading virus will be a persistent foe to economic reopening helped lead the Dow Jones Industrial Average down 710.16 points on Wednesday, its worst performance since June 11.

Political leaders also signaled concerns. In New York, Gov. Andrew Cuomo announced Wednesday that the state would delay reopening shopping malls, movie theaters and gyms, while the State Department of Health reviews more data about indoor viral transmission. In some areas of upstate New York, such facilities were set to reopen Friday.

New York, New Jersey and Connecticut rolled out self-quarantining requirements that would affect people traveling from Texas, Florida and Arizona, which have seen many more positive coronavirus tests and hospitalizations in recent days. Travelers from those states, among others, are now required to self-quarantine for 14 days.

"We have to make sure the virus doesn't come in on a plane again," Gov. Cuomo, a Democrat, said Wednesday. "Learned that lesson. Been there done that."

Some executives and front-line workers are pushing government officials to require masks. Nevada Gov. Steve Sisolak ordered face masks to be worn by everyone in public starting Friday, including gamblers inside Las Vegas Strip casinos. Las Vegas casino workers earlier this week called on Nevada's state leaders to require casino customers to wear masks.

Use of masks has been encouraged but remains optional since Nevada allowed gambling to resume June 4.

In Texas, one of the earliest states to reopen, several large companies have delayed plans to recall employees, while smaller companies that have restarted sooner are facing myriad financial and physical challenges.

On Monday, Gov. Greg Abbott of Texas said that Covid-19 cases there were spreading at an "unacceptable rate" and "must be corralled."

When Mr. Abbott announced salons were among the businesses that could partly reopen in early May, Tessa Holzworth's phone blew up. The owner of Blow Out Salon & Spa in Clear Lake, a suburb of Houston, received hundreds of text messages from clients after a seven-week closure. Ms. Holzworth and her staff wore masks and hired cleaners to disinfect the shop, and took clients into the shop one at a time, which often meant working long days. In a matter of weeks, though, she and four others tested positive for the virus, forcing her to close the salon again.

"We followed all the rules, followed all the regulations, and it still happened to us," Ms. Holzworth said. She reopened Tuesday and now requires all clients wear a mask. "I'm very happy Texas is reopening again," she said. "I know we can't all live in a little bubble forever."

At LyondellBasell Industries, a chemical company with plants and offices in the Houston area, a trickle of workers started coming back to work in May, said Kim Foley, LyondellBasell's vice president of health, safety and environment. The company has deemed it safe because people, wearing face shields, take employees' temperatures when they arrive and, at the end of the day, workers have to answer questions about whom they have had close contact with before they leave. The office never has more than 20% of capacity filled at any given time, and if anybody tests positive, those surveys would help trace potential exposure back to other workers so they could self-quarantine, she said.

"Our facilities, at our current occupancy rate, are safer than being in the general public right now," Ms. Foley said. Of the rising new cases in the Houston area, she said, "What we're seeing now is a result of people not exhibiting the right behaviors, such as not wearing masks nor practicing social distancing, and thinking that this pandemic is behind us."

After Aster Kebreab reopened her day-care center in Houston with the help of a small-business loan, she was hopeful the worst of the pandemic fallout was behind her. Now she may have to lay off her staff for the second time in a few months.

The number of children at her business, A to Z Preschool/Daycare Center, has dwindled from 45 before the lockdowns to about 16, and her loan money has run out, she said. She said she kept her doors open for the sake of essential workers, adding that following the Centers for Disease Control and Prevention's strict protocols has made it feel like two jobs.

"The hardest part right now is how to handle it next week," she said. "Am I going to afford to pay my mortgage and my employees? I don't think so. But I don't want to close it."

In Austin, Anne Rutt says she is committed to keeping her boutiques open even though most days it costs her more than she brings in. Customers at Triple Z Threadz, Limbo Jewelry and Little Limbo are required to wear masks and sanitize their hands before entering, which most are happy to do. A few have resisted, prompting Ms. Rutt to post signs at the front door reading, "Don't be a Karen, Wear Your Mask."

"You have to make them laugh and call them out without calling them out, " she said.

Business in the trendy area south of downtown was starting to pick back up, but many entrepreneurs there still fear it won't be enough to save quirky local stores in a part of town with high rents. Ms. Rutt said her sales are down about 80%.

"Austinites are staying home, and these are all tourists," she said.

--Elizabeth Findell contributed to this article.

Write to Collin Eaton at collin.eaton@wsj.com