You need to store the high score in some kind of shared persistent storage, either one you provide yourself or one provided by the portal hosting your game (if you are going that route, you'll want to consult their developer documentation; for example Kongregate's documentation on their leaderboards et cetera is here).
If you provide this yourself, you typically use a file or a database. There are projects, such as assql, that provide a mechanism to connect to SQL storage (in this case, MySQL) from ActionScript. You can also use ActionScript's methods for working with external data to hit a URL that is backed by a simply PHP script on your server which talks to a database on the server and recovers the high score. There are a surprising number of tutorials and posts about this scattered around the web, for example, this one.
The advantage to the PHP-based approach is that it puts all your database access on the server, rather than connecting to a DB directly from the client, which helps isolate you from a whole host of problems. You will of course want to further lock this down and make sure that users cannot submit bogus high scores simply by hitting, for example