The attack wounded an additional 200 recruits at the military camp in the southern province of Shabwa, the provincial government there said.
The rebels acknowledged targeting the camp.
The Iran-aligned Houthi rebels have put their growing military capabilities on display on multiple fronts in recent weeks, repeatedly striking Saudi Arabia's main international airport hundreds of kilometres away and seizing territory inside Yemen near a key shipping choke point.
In a sign that the Houthi attacks are unsettling Saudi society, the kingdom on Monday cancelled a global investment conference scheduled for next week and extended an order for schools in and around the capital Riyadh to teach students remotely.
The provincial government in Shabwa said the death toll could rise from Monday's attack, which it said "targeted new recruits in the training and preparation phase and is considered an extremely critical military escalation".
It called on Yemen's president, the Saudi-led coalition and the government to expedite air defence and early warning systems to reinforce Shabwa's capabilities.
The head of Yemen's ruling Presidential Leadership Council, Rashad al-Alimi, promised a "firm response".
A Houthi military spokesman said the camp was targeted because it is operated by Saudi officers and that some of them were killed in the attack, although he provided no evidence.
The Saudi government did not immediately comment.
A group of fighters aligned with the government said Monday its fighters repelled a Houthi offensive after hours of fighting in southwestern Yemen.
Separately, the local authorities in Yemen's Marib province said the Houthi rebels fired ballistic missiles at the densely populated city of Marib, which hosts large numbers of people displaced by the conflict.
The Houthis, a Shi'ite religious and political movement rooted in northern Yemen, are increasingly attacking Saudi Arabia because it supports Yemen's internationally recognised government in the civil war.
Saudi Arabia joined the war in 2015 after the rebels seized much of northern Yemen, including the capital Sanaa.
On Monday, Houthi rebels claimed to have carried out six attacks targeting oil infrastructure, airports and military sites across Saudi Arabia but the group did not specify when the attacks took place.
The Saudi Civil Defence issued danger alerts but no immediate reports of casualties or damage.
