For services it is better to use some kind of database connection management. Usually you can adapt Open/Close new connection on per-request basis. Note, most likely you will be working with logical connections and connection pool. Those are helpful to significantly reduce load to open physical connection. Physical connection is created without your direct control and is really heavy weight operation.
Do not put connection in a shared static variable, because connection is typically a disposable resource, which means you must dispose it. If something goes wrong and your connection to the database is corrupted then all your subsequent calls are deemed in doom.