I used static  connections to make sure there is
only one connection  to the database  in the whole
application, shared between the client classes. But
never in a static initializer, I use instead a static
wrapper method to access it (getConnection() or
something), so this method takes the work of, in the
first call, make the connection, handle  connection
errors, etc and in the following calls to check if the
connection is still valid and make again the
connection if it's not alive.
Remember that static initializers are not executed
when the application  starts. They're executed when the
class is loaded. So if the class with the static
initializer is not used, it will not be loaded,
neither initialized. Therefore, your static
initializer will be executed in the first reference to
the class (somebody tell me if I'm wrong).

another thing to consider:
some database  servers do not allow continuous connections - you'll have
to suss that one out.
if you can have continuous connections, perhaps use a lazy
initialisation that is run the first time a connection  is needed.

I dont think there is any big advantage of using
static intializer block for connection  if you are
using it only once.It will give you Connection object
just before the application  is started which you can
use later in your code.
According to me if you are using the database  heavily
then you should consider using connection pool from
where you can get connection object  as and when
required.It also reduces the overhead of opening and
closing the connection everytime you are using it.