Stalin's quote seems to originate from the 18th century quote which seems to apply as much today as it did then:
One Murder made a Villain, Millions a Hero. — Princes were privileg’d To kill, and numbers sanctified the crime. - 1759, Beilby Porteus, England
Saudi Arabia fears Iran acquiring another political control point - one on their border at that - in the Middle East and increasingly surrounding Saudi. So they're willing to scorch the earth to stop that.
Also, in terms of territory, with Iran + Persian Gulf, and Yemen + Bab-el-Mandeb straight, you can choke off Saudi Arabia's access to the Arabian Sea. That could heavily cripple their global oil shipping if it came to war, or at least cause a big mess. So Saudi Arabia views having a puppet and or friendly regime in Yemen, to be strategically vital. Saudi has wanted functional pipeline + port access in central Yemen for a long time, to get to the Gulf of Aden, to be able to bypass Iran's territorial waters just in case.