CARACAS, Venezuela—A U.S. pressure campaign to topple President Nicolás Maduro is stranding Venezuelans around the world, cutting off air links to one of the planet’s least-connected countries and compounding the misery of a beleaguered population.

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The fallout has been personal and immediate: holidays canceled, families unable to bring medicine to aging relatives, and cash-strapped Venezuelans marooned abroad.

President Trump said on Nov. 29 that the airspace should be considered closed after the Federal Aviation Administration warned operators of “a potentially hazardous situation” due to a massive military buildup near Venezuela. Since then, more than a dozen international air carriers have stopped serving the South American nation. A recent near-collision between a civilian jet and a U.S. military plane near Venezuelan waters further heightened concern.

“What airlines want to make sure is that we don’t have incidents over an airspace where an airplane is attacked because it was mistaken for a military carrier,” said Peter Cerdá, vice president for the Americas at IATA.

Reynaldo Goitía, frontman for the popular Venezuelan rock band Tomates Fritos, found himself stranded in Madrid after the Dec. 5 return flight he and his girlfriend were scheduled to take was canceled. Goitía, better known by his stage name Boston Rex, said he was determined not to use up his earnings from a tour for Spain’s large Venezuelan diaspora community.

That meant eating mostly fast food for days and sleeping on the floor of his friend’s office on a used mattress he bought for 80 euros, equivalent to $94. With no shower, he washed in a bathroom sink.

“When we saw the news that we couldn’t go back, we had to put ourselves in savings mode—a bit like putting your phone on low-battery mode,” Goitía said, before he embarked on a costly detour through Barbados to reunite with his daughter in Lechería, Venezuela, in time for Christmas. “You can’t do anything about it. You just feel powerless.”

Severing Venezuela’s air links adds to measures the U.S. is taking to isolate Maduro’s government, including intercepting oil tankers and imposing a partial blockade. The moves raise the risks of retaliation or a miscalculation that could push Washington and Caracas into direct conflict.

“Every time there’s some geopolitical problem, it’s always this industry that’s first to take the hit,” said Rodolfo Ruiz, managing partner at Ruiz & Partners, a Caracas-based law firm that works with airlines and insurers. “And, inevitably, it’s always the passengers, the average folks, who pay the biggest price.”

Alejandra Acuña, a Venezuelan marketing agent in Spain, said she had bought tickets to Caracas months earlier, hoping to introduce her partner to the homeland she left a decade ago. Her cousins had planned to fly over from Colombia for a family reunion, with costly medications for an elderly relative with Parkinson’s disease. Now, the trip is off and the family lost deposits for their trip to Venezuela’s Margarita Island.

“It’s unfortunate, we had to tell my parents at the last minute that we’re not going to make it,” Acuña said. “They said it’s probably better we just don’t come.”

It wasn’t always like this for Venezuelan travelers. In the 1970s, Caracas was one of the first destinations for Air France’s high-speed Concorde jets, a marker of a rising nation. From the 1990s through the mid-2010s, Venezuela was one of the highest-yielding aviation markets in the world, the IATA’s Cerdá said. An oil boom let Maduro’s predecessor, Hugo Chávez, subsidize travel. An overvalued bolivar currency turned Venezuelans into frequent fliers and big spenders abroad.

After Maduro took office in 2013, the economy began a downward spiral, with waves of civil unrest. Currency controls that barred companies from repatriating earnings forced airlines to write off billions of dollars. Carriers, from Delta to Germany’s Lufthansa, pulled out.

Travelers used to splurging in Miami and Paris suddenly had few options. U.S.-sanctioned state carrier Conviasa flew to a dwindling number of allies in Latin America and ran seldom-used routes to Moscow, Damascus and Tehran.

In the last couple of years, Venezuela averaged about 15,000 passengers flying in and out of the country weekly, according to IATA, mostly from Spain, Portugal, Panama and Colombia. Traffic has now plunged to about 1,000 to 2,000 a week.

The flight disruptions are also upending essential cargo such as pharmaceutical products and perishable goods, as well as aircraft parts that Venezuelan carriers need to maintain their fleets, said Cerdá.

With the U.S. deployment of warships to the southern Caribbean, safety has become a top concern, Cerdá said. On Dec. 12, a JetBlue plane leaving from Curaçao—40 miles north of Venezuela—and heading to New York reported a near midair collision with a U.S. Air Force refueling jet.

Industry insiders said the incident may have been due to GPS and radar jamming, complicating navigation as both the U.S. and Venezuela beef up military hardware.

Maduro says the airspace troubles are part of a psychological war to seize Venezuela’s oil wealth. Venezuelan authorities ordered national airlines to keep flying and stripped operating licenses from some foreign carriers that suspended flights.

“You guys can keep your planes and we’ll keep our dignity,” Interior Minister Diosdado Cabello said, denouncing airlines for cooperating with the U.S.

Vice President Delcy Rodríguez recently said the government was working to find alternative routes with Venezuelan carriers.

Some 40,000 people have seen their travel plans affected for the month of December, the peak season for the industry, said Gregory Barrios, an aviation expert who runs a travel agency out of the Venezuelan state of Aragua.

Authorities have been able to increase flight frequencies to a few countries in the region, mostly to neighboring Colombia. But Barrios said he was concerned by how quickly international carriers acted to cut Venezuela off after the U.S. warning.

“It’s a very worrisome dynamic,” Barrios said.

Write to Kejal Vyas at kejal.vyas@wsj.com