Sea snakes are not particularly hard to keep and various species have been kept long-term. This isn't a recent development, but has been the case since the 1990s at least. However, only sea kraits have been seen with some regularity in aquariums in Europe and North America, probably because these coastal species are easier to source than the true sea snakes. True sea snakes have only very rarely been kept outside their native Asia and Australia. As long as they are healthy when entering captivity, they are not particularly prone to disease and readily feed on dead fish (there's also videos on youtube: "Sea Snakes feeding at Reef HQ Aquarium" and "Sea snake feeding at Ocean Park Aquarium, Shark Bay WA"). When well-fed they evidently tend to ignore most other fish in their exhibit. For example, a wide range of small to medium sized fish (squirrelfish, wrasse, surgeonfish, angelfish, butterflyfish, moorish idol, damselfish, maskray, etc) co-inhabit the olive sea snake exhibit at Blue Planet Aquarium (Denmark); most of these have been together for a very long time without issues. Reef HQ (Australia) has successfully kept sea snakes with lionfish and stonefish for a long time. Aquarium of the Pacific (USA) and Berlin (Germany) have kept a range of small fish with their sea kraits, but I'm unaware of their "durability", i.e., if fish remain long-term or frequently switch, suggesting predation by the sea snake. However, mixing should still be done with care. Several few years ago an aquarium in Australia lost a sea snake... when attacked and eaten by a pufferfish. Since many sea snakes feed heavily on eel, especially morays, in the wild, they presumably should never be mixed. I've seen a photo from Aquarium des Lagons (New Caledonia) where a sea krait and sea turtle share a tank, but suspect this is a very risky mix.
The world's first captive breeding of a sea snake, the olive, happened early this year at Reef HQ and a few months later it was bred at Blue Planet. I'm not aware of captive breeding of sea kraits, but I may have missed it. They obviously require a land section, being less "sea" than the true sea snakes. Yellow-bellied sea snake doesn't need an ocean tank, as one might think due to its pelagic behavior, but has done fine in normal tanks. A problem with this species is that most that have ended up in aquariums were stranded, unhealthy individuals and under those circumstances they don't live long (the same for Shedd's confiscated yellow-bellied that were in a bad state when received by the aquarium). Such sick, stranded individuals have been transferred to aquariums in USA (California, Hawaii), New Zealand, Australia and probably elsewhere. Only in Australia do I know of an aquarium, Melbourne, that kept it for a longer period. I don't know what happened to the yellow-bellied sea snake that was turned in to an ordinary aquarium store (!) in Sydney in 2015, but I know they informed the authorities about it and guess it was forwarded to a public aquarium. However, there are two issues with this species compared to the other typical sea snakes: It is an ambush predator (other true sea snakes are active hunters) and feeding has to be adapted for this. Secondly, studies in the last few years suggest that it may require access to freshwater for drinking.
The issue of keeping ultra-venomous snakes obviously hasn't changed and they require special security/handling techniques.