Government officials in the New York City area urged calm following an Ebola patient’s death in Texas, saying they were taking necessary steps to protect the region against the virus.

Their assurances Thursday came as workers who clean airplane cabins and bathrooms at La Guardia Airport went on a one-day walkout to protest what they said were inadequate protections from potential exposure to a disease whose recent spread has set off world-wide alarm.

The officials’ statements and airport workers’ concerns reflect growing worry over whether countries with advanced health-care systems could effectively prevent and contain an outbreak that so far has been concentrated in West Africa.

Mayor Bill de Blasio said previous catastrophes had prepared New York City to face the virus, which claimed its first known U.S. victim Wednesday since the recent outbreak.

“The idea of having to be ready for something very challenging is not new to us—it’s what we do as New Yorkers,” Mr. de Blasio said Thursday after attending an Ebola preparedness meeting alongside health and emergency-management officials. “We’ve faced terrorism. We’ve faced natural disasters. We now face the possibility of being affected by a pandemic. But the fact is, all of those other moments prepared us to handle whatever is thrown at us.”