This is a really specific unique inquiry. By general stability logic you would think that Iran would benefit and therefore desire a stable Yemen. However based off their actions that seems to not be the case. The southern secessionist movement has been in play since the unification of southern and northern Yemen. Southern Yemen being historically Communist and before that, under British ‘control’. It is unique to consider the religious regime of Iran would be supportive of a fundamentally atheist movement (communist) but I can see the intricacies. There have long been accusations of Iran meddling in Yemeni internal affairs. Knowing that Iran is fairly good at subtle subterfuge it is not surprising that this has been difficult to prove for certain.
Recently Yemen President came out and publicly stated that Iran has been meddling in Yemeni internal affairs particularly regarding support of the Southern Secessionist movement. However there is also a large amount of speculation (with minimal proof) that Iran directly supports the Houthi movement (partially as a proxy war against the Saudi supported Salafists). I have seen some minimal reporting (validation I’m not sure of) claiming that Iran has IRGC operatives in Yemen (Sa’ada region) training and funding the Houthi movement. Though by their very nature these types of clandestine operations are hard to prove. There are however, a ridiculous number of weapons shipments intercepted on a regular basis that point to Iran. These shipments are presumably directed towards the Houthis in Sa’ada and/or the Southern movement. Most people in Yemen I have spoken to reject the notion of Iranian support to the Houthis. Though most of these people were Shia (ergo Houthi affiliated to some degree).
Considering Iran is looking to expand commercially via infrastructure into the Arabian Peninsula, I believe Iran wants a stable Yemen in order to extract markets and potentially exploit various natural resources. However they also want an Iran-friendly region with power-pockets of Shia (this would explain why they support the Houthis and Secessionists. Ideally they would support the complete dismantling of the current pro-US regime in order to replace it with a preferred administration. However, the chances of that happening is slim to known so it seems Iran is left to just endlessly meddle in their affairs in a childish manner.
If there is any true Iranian strategy that is conceivably possible it would be to destroy the efforts by the GCC with regards to a peaceful transition and development (the mostly Sunni/Saudi backed GCC is heavily involved in Yemen’s NDC and Saleh Regime transition) in order to create a power vacuum and alliance shift to Iran and away from GCC/Saudi affiliation. If Iran can show that GCC efforts are failed efforts that they probably believe they can grab some power in the country and particularly have a foothold in the Arabian Peninsula.
In reality, Iran is never going to “take over” Yemen by proxy or otherwise, the best option they have from a pure realist perspective is to support a Houthi and Southern movement effort to have their own autonomous or semi-autonomous areas that Iran can use as proxy-forward operating bases on the Peninsula. The way this would be done is to have these two factions/movements cause enough trouble in the peace process to the point that the UN/GCC/US coalition provides concessions to these movements. This has already been done to a great degree with the Houthis. There has been a good amount of commentary on how before the Saleh regime the Houthis were nobody and now they have a seat at the table – one that I actually sat at during the final moments of the NDC. They were rather furious that an “American Researcher” was sitting in on their meeting that was to decide the future of Yemen from a Houthi perspective.
In the end, Iran would love to take over Yemen, but they cannot, they can merely push their hands into the peace process to extract a degree of proxy power via the Houthis (and lesser so the Southern movement). As per whether they support the Houthis it takes only a walk through a Houthi neighborhood (I lived in one) where you see the Houthi motto banners everywhere saying effectively “Death to America, Death to Israel” – and we all know whose motto that is…Iran’s.