The US has stepped up support for a Saudi-led coalition against the Houthis, who drove the Yemeni president out of the country.
Tehran has denied accusations it is backing the rebels.
However, it reportedly despatched navy vessels to the Gulf of Aden, off Yemen, on Wednesday.
Navy commander Rear Adm Habibollah Sayyari told state media the move was made with the aim of "safeguarding naval routes for vessels in the region".
Speaking to PBS Newshour, Mr Kerry said it was clear Iran was aiding the Houthis: "There are obviously supplies that have been coming from Iran. There are a number of flights every single week that have been flying in."
"Iran needs to recognise that the US is not going to stand by while the region is destabilised or while people engage in overt warfare across lines, international boundaries and other countries," he added.
Fighting has intensified in the southern Yemeni city of Aden between the Houthis and militiamen loyal to President Abdrabbuh Mansour Hadi.
Meanwhile, warplanes from the Saudi-led coalition which backs Mr Hadi have been bombing rebel targets in the country`s north.
`Aerial bombardment`
The Iranian Foreign Minister, Mohammad Javad Zarif, has proposed a four-step solution to ending the crisis, including a ceasefire and formation of a broad-based government.
"The people of Yemen should not have to face aerial bombardment," he said while visiting Pakistan, which is currently debating whether to join the Saudi-led operation.
At least 22 people were reportedly killed on Wednesday by shelling in two residential districts in Aden.
After days of street fighting, conditions for civilians have been described as catastrophic by aid groups.
On Wednesday, boats carrying aid managed to arrive in Aden for the first time since the Saudi-led air strikes began.
A ship carrying 1.7 tonnes of medical aid for Medecins Sans Frontieres, and another Red Cross boat carrying medical supplies and staff, docked at the port city.
The World Health Organization (WHO) has said that at least 560 people, including 76 children, were killed in Yemen between 19 March and 4 April. Another 1,700 people were wounded and 100,000 fled their homes.
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