Good technique for connections with PostgreSQL
PostgreSQL supports connections pooling (pool size is customizable) so the common pattern:
using (NpgsqlConnection conn = new NpgsqlConnection(...)){...}should be the better choice.
In my opinion you should open a connection at the moment you need it, and close it right after it. This will prevent a lot of connections on the server to be kept alive.
In my experience, opening a connection doesn't take that much time (a few milliseconds, usually a fraction of your execution time), so you don't have to worry too much.