I've only spent a very limited amount of time in Mobile, but from what I can tell, the options are limited.
Cooper Riverside Park downtown would offer views of the vessels at the Austal USA shipyard, and any passing traffic. Unfortunately, it's upriver of most of the deepwater port, so I would assume it's mostly tug and barge traffic at that point.
Distant views of the ships coming and going look like they would be available from the boardwalk at Arlington Park near Mobile Downtown Airport, and from Fort Gaines, Fort Morgan or the Mobile Bay Ferry out at the mouth of the estuary, though the viewing distance is probably well over a mile for those locations.
Out at Bayou la Batre, there is a boat launch south off of Shell Belt Road, at the mouth of the harbor, that probably offers a great view of the fishing boats coming and going.
From what I can find, perhaps the best option is the local 2 hour harbor cruise offered on Sundays. It costs $20, and they ask for reservations in advance: http://5rds.com/the-secretly-awesome-harbor-tour/
Hope this helps,