For screenplay documents, it is recommended to use a screenplay-specific Courier font. These can be found at www.fadeinpro.com on the Download page under "Extras". Courier Screenplay and Courier Prime are both available and free for download.
Note, however, that these fonts primarily support Latin alphabets. If you are writing in a language that uses a different alphabet, such as Greek or Russian (or other Cyrillic languages), etc., it may be necessary to use a font with broader character/glyph coverage, such as Courier New or a language-specific Courier. (There exist, for instance, separate Courier Greek and Courier Cyrillic fonts.)
If you are writing in a language such as Chinese, Japanese, Korean, Thai, Vietnamese, Hindi (or other Devangari languages), etc., you should use a font that includes the appropriate character set.
Trying to write in a language with a font that does not properly support that language may result in words appearing incorrectly, blank spaces, or white rectangles.
While Courier New is a highly available font and has excellent glyph coverage, it is often considered too thin/light for typical screenplay usage. Using one of the screenplay Couriers is generally preferable.
(Note for Windows users: it is recommended not to use the default plain "Courier" font, for these reasons.)