Gulf

Capitalize when referring to the Gulf of Mexico:

Other gulfs are always lowercase except in a place-name:

Gulf of Mexico, the Gulf

Gulf Coast (U.S. region)

Gulf Coast states

Gulf current

the Persian Gulf

the gulf

Persian Gulf states

the gulf coast

Gulf of Corinth

Gulf states (for Arab states of the gulf)

the Gulf region (U.S.)

Gulf Stream

Gulf waters

Note that although the historical and internationally accepted designation is Persian Gulf,

many Arab countries use Arabian Gulf.

Where scale permits, National Geographic maps  include a note about Arabian Gulf.

If Arabian Gulf is used in text, it should be explained.