"Our brave Warfighters are right now carrying out aerial attacks on the terrorists' bases, leaders, and missile defenses to protect American shipping, air, and naval assets, and to restore Navigational Freedom," Trump said in a social media post.
"No terrorist force will stop American commercial and naval vessels from freely sailing the Waterways of the World."
He also warned Iran to stop supporting the rebel group, promising to hold the country "fully accountable" for the actions of its proxy.
The Houthis reported a series of explosions in their territory Saturday evening.
Images circulating online show plumes of black smoke over the area of the Sanaa airport complex, which includes a sprawling military facility.
The extent of the damage was not yet clear.
The air strikes come a few days after the Houthis said they would resume attacks on Israeli vessels sailing in waters off Yemen in response to Israel's blockade on Gaza. There have been no Houthi attacks reported since then.
The United States, Israel and Britain have previously hit Houthi-held areas in Yemen.
Israel's military declined to comment.
"These relentless assaults have cost the US and World Economy many BILLIONS of Dollars while, at the same time, putting innocent lives at risk," Trump said.
