SQL MATCH AGAINST incomplete string with space
If you want to use Full-Text Search, then in order to get an exact phrase match it would seem that doing a Boolean Full-Text Search would be the way to go, enclosing the term New York in double-quotes, hence:

SELECT * FROM places WHERE MATCH (field) AGAINST ('"New York"' IN BOOLEAN MODE);


But, if you are looking to do a "contains" type of search where the user-entered value is a literal fragment (i.e. what they entered needs to match as-is, but can also be a part of a larger string and is not necessarily the entire match), then perhaps doing a regular LIKE '%{user_input}%' would be best.