They have been around for decades and have survived the changing landscape of the financial services sector since the 1970s. Past success is no guarantee for the future, though.
There is no deposit insurance in the Cayman Islands. However — albeit at the risk of forgetting something — there have been no bank failures except for example Silicon Valley Bank's branch in the Caymans.
Altajir's internet banking is a bit vintage, IIRC. It's all very simple and barebones.