The Houthis' military spokesman, Brigadier General Yahya Saree, said on social media on Wednesday that the rebels launched ballistic missiles and drones in the attacks targeting King Khalid International Airport in the capital, Riyadh, but did not provide evidence.
Flights to and from the airport were briefly disrupted, according to Flightradar24, a tracking platform.
Saree said Abha airport, a frequent target of Houthi drones and missiles, was targeted.
The rebels also claimed to have hit an air base in the city of Khamis Mushait and two military bases Najran and Asir.
All are in southwestern Saudi Arabia and near the border with Yemen.
The spokesperson for the Saudi-led coalition fighting the Houthis, Major General Turki al-Malki, said they intercepted and destroyed a ballistic missile north of Riyadh.
Saudi Arabia and allied Yemeni forces in recent days have launched a major ground and air offensive against the Houthis in an attempt to retake territory that the rebels seized about a month ago along Yemen's western Red Sea coast and on the Bab el-Mandeb Strait.
Those positions gave the rebels better surveillance of shipping as they pursue a blockade of Saudi vessels.
On Monday, officials said forces of the Saudi-led coalition fighting in Yemen recaptured several key areas around the strait.
The fighting has since then centred in and around the crucial town of Dhubab, some 50km south of the port city of Mokha, according to both the Houthis and the Yemeni government.
Clashes were also reported in the district of Waziayah in southwestern Taiz province, which links Yemen's mountainous highlands to the Red Sea, and overlooks roads to the Bab el-Mandeb Strait.
The Houthis also said they launched a missile strike early on Wednesday against Yemen's port city of Aden, the seat of the internationally recognised government.
The rebels hit a military camp close to the Aden airport with ballistic missiles, Saree said.
Security authorities in Aden said the Houthis targeted the airport with two missiles, with no casualties.
The airport is close to the seat of government.
"The targeting of this vital civilian facility represents a direct threat to the safety of civilians, passengers, employees and flight crews," the Aden police said in a statement.
The attack on the Aden airport threatened "a humanitarian disaster", as one missile hit close to the runway just before the scheduled landing of a flight from Cairo, the human rights ministry said.
"The Houthis believe that they can maximise Saudi economic costs and compel Saudi Arabia to surrender to Houthi demands by striking civilian infrastructure, such as oil facilities and airports," the Institute for the Study of War, a Washington-based think tank, wrote in an analysis.
