you go into the aquarium place, and he will be there.
a guy will ask u if u can deliver Devon goods to captain stern..captain stern is found in the museum of slateport(specifically 2nd floor).
Head to slateport city and go to the SS tidal captain stern will be nearby.
You deliver Devon's Goods to Captain Stern who is in Slateport City's museum.
Captain Stern is on the top floor of the museum in Slateport City.
Go back to Slateport town. Inside the ferry building, you will find captain stern and the submarine being robbed by Team Aqua.
after you beat the elite for the ferry is complete and captain stern is the captain of the ferry (i think check to find out)
he will be at slatport city after you find the team magmas hideout having a interview
The captain went to the stern of the ship to see if anyone was following them.
Hospital
In the Slateport museum.
Cool story bro.
While in Lilycove City, you can find Captain Stern in that museum place where Team Magma/Aqua are standing by. I forgot how to get rid of them, but once you do, you can find Captain Stern in that museum place I mentioned.