The Houthis are more a nuisance than anything else. The U.S. has no compelling need to be the guarantor of Red Sea shipping, although a lot of countries and private companies would like it to be. From Eugene Gholz at antiwar.com:
Last week’s US/British strike against Houthi military targets in Yemen escalated operations to protect shipping in the Red Sea from an expensive and wasteful defensive project to an offensive that could draw the United States further into the Yemeni civil war. Even Saudi Arabia, which has been fighting the Houthis for years with support from the United States and with little success, did not favor the US/British strike, stressing the need for “restraint and avoiding escalation.”
The US‐led coalition, Operation Prosperity Guardian, which began December 18, costs the United States too much. Deploying Navy ships and aircraft to the Red Sea costs US taxpayers for fuel and munitions and other consumable items (e.g., radar decoys that might be deployed as part of the operations), and exposes those US ships and aircraft to greater (though still modest) risks of combat.
Some of the weapons against the Houthis are inexpensive by US standards (say, rounds from US destroyers’ 5‑inch guns or helicopter‐launched rockets that might hit a Houthi speedboat or unmanned surface vehicle trying to attack a ship). But even those cost a lot compared to the cost that the Houthis (or their patrons) pay to mount the attacks, creating an unfavorable financial tradeoff for the United States.
Other US weapons likely engaged are much more expensive: Standard missiles that engage anti‐ship ballistic missiles cost millions of dollars each, and even the cheaper air‐to‐air missiles potentially used against Houthi anti‐ship cruise missiles (or, worse yet, drones) cost hundreds of thousands or even a million dollars each. (Though the Department of Defense has not confirmed the specific munitions used to thwart Houthi attacks, we know the types of weapons that the US forces generally use to defend against missile and drone attacks, whether the defensive weapons are launched from aircraft or surface ships.)
The indispensable ally demands it.
Once saw a comment about the scene in Mad Max III with the mentally defective brawler controlled by the demanding little man as a metaphor for the relationship with our extra most bestest ally.
Just read about Spain and some others dropping out of the bomb one of the poorest countries on earth Yemen.
The Spanish had a wymyn minister of WAR, spicy!