The attack took place near the border between Saudi Arabia and Yemen and came as talks between Riyadh and the Iran-backed Houthi rebels have been gaining traction in recent weeks.
“This unprovoked attack threatens the longest period of calm since the war in Yemen began,” US State Department spokesman Matthew Miller said.
Washington has expressed solidarity with Bahrain and Saudi Arabia after the drone strike.
“We wish those injured in this unprovoked and unacceptable attack a speedy recovery and return to duty,” Mr Miller said.
The war, which has been raging since September 2014, has been in a relatively stable and quiet period after a six-month, UN-backed ceasefire that went into effect in April 2022.
While the ceasefire has long expired, fighting has remained quiet, with the Houthis and Saudi Arabia actively engaging in talks.
A Houthi delegation was in Riyadh last week, the first such occurrence since the war began.
“We have worked tirelessly with our partners to de-escalate, secure a truce and incentivise the parties to launch a Yemeni-Yemeni peace process,” Mr Miller said.
Yemen was a major talking point at the UN General Assembly this year, with Washington taking part in several high-level meetings on the subject.
“The Secretary [of State, Antony Blinken] discussed peace in Yemen in a number of his engagements with counterparts in the region last week during the United Nations General Assembly, and emphasised that only a Yemeni-Yemeni political agreement can durably resolve the conflict,” Mr Miller said.