I don't know New York Times style, but this was on the cover of the New York Times Magazine:
with guidance from a U.S. adviser who in the 80's commanded the Special Forces
Both The Associated Press Stylebook and The Chicago Manual of Style support these (more logical) versions instead:
In the article itself, 1980's is used several times. If the editors were to replace the missing digits with an apostrophe, '80's would look awfully weird, which is one of the reasons why the other style is preferred. Also, it looks like a possessive, and if you ever wanted to use a possessive form of it, where would you go from there? '80's'?