But the most important effects of a destabilized Yemen could be on the Saudis just to the north.
The Houthi rebels present a problem to the kingdom, as the newly crowned leadership in Riyadh worries about being flanked on several fronts by Shiite Iran, Cordesman said.
"Clearly, we have seen the hands of Iran infiltrating that country through its blatant and open support of the Houthis there," Saudi royal and investor Prince Alwaleed Bin Talal told CNBC last week.
He called it an "unfortunate situation" as Yemen's political vacuum could become a "seat of trouble" in the region.
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Beyond concerns related to Iran, any humanitarian crisis along the Saudis' borders could make the authoritarian regime in Riyadh nervous about stability on the peninsula, experts said. The Saudis have watched one allied Arab state after another become destabilized over the last few years, and are now watching that chaos migrate to a country that shares their peninsula.
For now, Saudi Arabia has largely taken a "wait-and-see" approach to Yemen, Grill said, adding that the country was even tapering off its support of the Hadi government in recent months as the tenuous political situation began to worry Riyadh more.
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"Saudi Arabia is still weighing their bets; they don't know how they're going to deal with it," she said, adding that the next test of the situation in Yemen will be if the rebels actually accept Hadi's resignation—many observers thought the Houthi only hoped to gain influence, not lead a government, she explained.
This state of affairs may not last, however, as Saudi Arabia's new leadership will likely take a more hawkish approach than under former King Abdullah, said John Kilduff, founder of Again Capital.
"While Abdullah was prone to exercise restraint—to a degree—and patience, I don't see [new Deputy Crown Prince Mohammed bin] Nayef doing that," Kilduff said.