The full answer

"With" sits exactly at Chicago Style's four-letter threshold, which lowercases prepositions of four letters or fewer. AP Style lowercases all prepositions, so "with" is also lowercase in journalism. NYT Style and Billboard Style capitalize it.

AP Style lowercases all prepositions regardless of length, so "with" is always lowercase in journalism under AP rules. AMA Style follows similar logic, keeping "with" lowercase across medical publications.

NYT Style capitalizes most prepositions including "with," producing headlines with more uniformly capitalized words. This is one of the visible differences between AP-style and NYT-style headline formatting.

"With" as the first word of a title is always capitalized. This is somewhat unusual for a preposition, but titles beginning with "With" — such as journal article subtitles or instructional phrases — follow the first-word rule.

"With" is exactly four letters long, which places it at the boundary of Chicago Style's preposition threshold. Chicago Style lowercases prepositions of four letters or fewer, so "with" is lowercase. A five-letter preposition like "about" would be capitalized in Chicago Style.