Yes, Stream is considered production-ready and safe for use on large sites. There has been some careful attention paid to accommodating for large numbers of users.
You also have the ability to set how many days Stream records should be kept before being automatically purged (the default is 90 days). This is done in the Stream > Settings > General screen.
One thing I might warn you of is failed login attempts. Stream will always track these for you, if you have a lot of users and no security plugin for deterring brute force attacks there is a chance these logs could grow quite rapidly. Please see this thread for more info.
But as with any plugin on a high-traffic site, especially one that creates new DB tables, you should test this on your staging server first as your own sanity check and to see how the plugin might benefit you before it goes to production.
Have a good one! If you have any issues at all, just report them here.