The Iran-allied Mouths have launched dozens of attacks on international shipping in the Red Sea region since November in solidarity with the Palestinians in the war between Israel and Hamas.
Thursday's attack marked their second direct hit on a merchant ship in two days, and the group said its campaign would continue until hostilities in the Gaza Strip end.
Three missiles struck the Verbena on Thursday, sparking a fire and damaging the ship, the United Kingdom Maritime Trade Operations (UKMTO) agency said.
While her crew fought the fire, an aircraft from the U.S. Navy's Philippine Sea warship medically evacuated the injured mariner to a partner force ship nearby for medical attention, U.S. Central Command said.
Reuters was not able immediately to contact the Verbena's Polish manager. The vessel, loaded with wood construction material, was sailing to Italy at the time of the attack, CENTCOM said. Security and military sources said the Verbena still has power and steering capabilities.
A day earlier, Yemen's Houthi forces took responsibility for small watercraft and missile attacks that left the Tutor, a Greek-owned cargo ship, taking in water and in need of rescue near Yemen's Red Sea port of Hodeidah.
The Philippines' Department of Foreign Affairs condemned the assault on the Tutor, which had Filipino crew on board.
Some media outlets have reported that one person died as a result of the attack on the Tutor. Greek shipping authorities said they had no confirmation of that. Manager Evalend Shipping has not responded to Reuters requests for comment.
The Houthi campaign in the Red Sea region has disrupted global shipping, cascading delays and costs through supply chains. The fighters have sunk one ship, seized another vessel and killed three seafarers in separate attacks.
"Operations will not stop unless the aggression stops and the siege on the Palestinian people in the Gaza Strip is lifted," the Houthis said on Thursday.
They claimed to also have directly hit two other ships. UKMTO said the master of one of those reported an explosion near the vessel that did not cause any damage or injury. Reuters was not able to immediately verify the other account.
While several near misses have been reported, "the data would also say (the Houthis) are getting more successful with direct hits," Joshua Hutchinson, managing director of intelligence and risk for British maritime security firm Ambrey, said in a post on LinkedIn.