The airport is disused but housed a major rebel base just inland from the coast road into the city from the south.
Aramco said its facilities in Jizan, in southwestern Saudi Arabia, were "intact" and operating normally, after pictures were posted on social media of a fire sparked by a missile attack.
The Houthi rebels, locked in a war against Yemen's Saudi-backed government, said they had fired a missile at the kingdom's southwestern province of Najran in a statement tweeted by their Al-Masirah television channel.
Saudi Arabia and its allies struck a day after Saleh's son vowed to lead a campaign against the Iran-aligned Houthis.
The head of the Arab League warned that Saleh's killing following the collapse of his uneasy alliance with the rebels threatens an "explosion in the security situation" in the war-ravaged country.
Yemen's Houthi rebels said Thursday they fired a ballistic missile at Saudi Arabia and hit a military target, in the second such attack this month, after threatening to retaliate over a crippling blockade.
The stark warning from World Health Organisation, Unicef and World Food Programme came 10 days after the Saudi-led military coalition shut down Yemen's sea and air ports.
Relief supplies into Yemen have been blocked since the coalition sealed off the country's borders following a missile attack by Yemen's Houthi rebels that was intercepted near Riyadh airport.