Back in the early days of the web, search engines such as Yahoo explicitly advertised how you could do boolean searches using keywords like 'AND' and 'OR.

Eventually things moved to a much simpler phrase based search. The search phrase was broken down to keywords and an algorithm would rank results based on how well it matched the keywords (basically) - like a fuzzy AND query between the keywords.

MySQL full text search works in the newer way, although you can also force a boolean search mode.

Composr supports both modes.

I want to know if nowadays anybody cares about, or expects, boolean search support.

For our site I direct people to the Google search, that just searches the google index the they have for our site. 
I cannot remember the last time I used the Boolean search.