You could just obtain the connection, as you said (I haven't checked the code, but it should be easy to check if it works or not, and under which circumstances).
However, placing this code in init() is plain wrong: init() is typically called exactly once, and most likely right after server start (or rather right after the portlet's deployment) - just like a servlet's init() would be. You'd have to implement this in a portlet's render phase, and if you try to read anything from your database but get an exception that hints at the database's unavailability: Catch that exception and show an alternative message. Liferay only shows the "portlet unavailable" when the render phase triggers any uncaught exception. If you catch it yourself, it's up to you to determine what you show.
I'd just access the database through servicebuilder, without mocking around with the underlying connection, and catch a resulting exception, rather than coding low level database stuff. You could even hide this in your LocalServiceImpl and implement an available() (or whatever smarter name you come up with) method there.