Cool.

I first heard a very similar line sang by the youngest child 'Wesley' in the show
Mr. Belvedere
in 1980s. 'Wesley' was in a play at school and he sang, "and I'm hardly ever sick at sea." It must be from the play that these other people are referring to. I never forgot that line so when I heard it in
Malice
I thought of that connection.

It means he never, ever panics.

He's never sick at sea = he never f-cks up a work assignment. Always delivers.
'facts are stupid things' - Ronald Reagan

He said he was certified at 7 different medical boards in New England I think he might be referencing the east coast in general, boats, team rowing, yachting, etc. Just a brag that he is steady under all circumstances where others would be woosey like after drinking all evening.

From "My Gallant Crew, Good Morning!"
Captain.
Though related to a peer,
I can hand, reef, and steer,
And ship a selvagee;
I am never known to quail
At the fury of a gale,
And I'm never, never sick at sea!
Chorus.
What, never?
Captain.
No, never!
Chorus.
What, never?
Captain.
Hardly ever!
Chorus.
He's hardly ever sick at sea!
Then give three cheers, and one cheer more,
For the hardy Captain of the Pinafore!
Then give three cheers, and one cheer more,
For the Captain of the Pinafore!

You are correct. Aaron Sorkin is one of the writers for Malace. His bio states: Many of his works contain references to the operas of Gilbert & Sullivan. In Malice (1993), the doctor played by Alec Baldwin boasts that he is "never, ever sick at sea", lyrics from "The HMS Pinafore".

It's a very weird line. I think youre right about Pinafore or other literary reference.

I think he was using a line from H.M.S. Pinafore. It was an Opera from the late 1800s.