|NASA photo: Bab al-Mandab with Perim Island|
Let's start with this: you and what navy?
In response to the far less lethal threat of Somali piracy, the United States, NATO, the European Union, and even Russia, China, and Japan, dispatched warships to assure freedom of navigation. Do you think they'd let the Strait be closed? If somebody fires on a ship from Perim Island, I think they'll get an up-close and personal visit from an Aegis Cruiser (if anything is still standing after the Predator strikes).
Even if the international warships in the Gulf and Indian Ocean were not in the neighborhood, the Egyptian, Saudi, and Israeli navies are sufficient, I suspect, to deter or meet any threat. It's not the days of Alfred Thayer Mahan or Teddy Roosevelt. Sea power is global, three dimensional, and rapidly deployable.
I'm reminded of my days writing on defense issues in the region in the early 1980s, when the Soviets had bases in South Yemen and Ethiopia, the US had bases in Oman and the French in Djibouti, and there was a lot of talk about Bab al-Mandab. Unlike the Houthis, the Soviets could have closed the Strait. (At the cost, of course, of starting World War III.),
Iran has from time to time threatened to close the Strait of Hormuz. And it has a navy that could credibly make the attempt. At the cost of war with the US, NATO, and probably loss of its own oil production facilities.
Though international traffic passes through territorial waters of Yemen and Djibouti (as Hormuz does between Iran and Oman and the UAE), these are international straits where the right of innocent passage (or "transit passage" as the International Convention on the Law of the Sea calls it) is guaranteed.
The closest thing to a closure of the Strait I know of was an Egyptian Navy blockade during the 1973 Arab-Israeli War, which was not aimed at the world's oil supply but at intercepting shipments bound for the Israeli port of Eilat. And at the time, the Suez Canal had been closed since 1967 anyway. And it was wartime, when the rules change.
Which brings me to another point. Blocking Bab al-Mandab might be possible with a serious naval force, which Yemen lacks. But is it really the most vulnerable point in the oil supply? Let's run some numbers:
Bab al-Mandab: Width 18-20 miles (16 miles between Perim Island (Yemeni) and Djibouti coast.
Hormuz: Width 21 miles, but with shipping channels only two miles wide in each direction, separated by a two-mile buffer.
Suez Canal: Width 673 feet.
SUMED Pipeline: two 42-inch pipes
If you want to block international tanker traffic, which choke point is chokiest?
Due to Arab-Israeli wars, the Suez Canal was closed for several months in 1956-57 and again from 1967-75. Sinking a few ships in the shallow canal can block it for weeks. If bad guys also attacked the SUMED (Suez-Mediterranean) pipeline (two 42-inch wide lines running from ‘Ayn Sukhna on the Red Sea to a terminal off Alexandria), not only would passage of Gulf and Yemeni oil to Europe be blocked, but so would Saudi oil from the terminal at Yanbu‘ and Sudanese oil from Port Sudan.
I really am less worried about the Houthis blocking Bab al-Mandab than I am about the instability in Sinai threatening the Canal and SUMED.