When adapting and updating a well known property in the public domain, it is a bit of a judgement call as to whether or not the original work needs a “based on” credit.
If it feels dishonest not to note it, note it. If not, it’s up to you.
For a case like West Side Story, the connection to Romeo and Juliet is obvious enough that no one will accuse the writers of stealing. Audiences may discuss it as “a musical Romeo and Juliette set in 1950s New York,” but as long as what they like about it is not solely the lovers-from-rival-families element, no one will look at the fact that it is an adaptation as a bad thing.
If you are adapting an obscure 14th century fable that is full of twists, however, and your modern sci-fi version is getting praise for its clever and original turns, it may ease your conscience to note the original story.
And of course, if it’s just a straight page-to-screen adaptation, you should certainly credit the original writer.
* [Is it risky to spec something in the public domain?](http://johnaugust.com/2007/public-domain-risk “Is it risky to spec something in the public domain?”)
* [Dead copyright holders, and being too young](http://johnaugust.com/2004/dead-copyright-holders-and-being-too-young “Dead copyright holders, and being too young”)
* [Why aren’t adaptations ok for competitions?](http://johnaugust.com/2009/adaptation-competition “Why aren’t adaptations ok for competitions?”)