The Sound Alerts browser source is the "media-player" that plays all the audio and displays the visual alerts of the Sounds Alerts that are played by your viewers. This source needs to be implemented in your streaming software like OBS in order to broadcast the sounds and the visuals to your Twitch stream, and also that you, as the streamer can hear the actual Sounds Alerts as well.
The reasons why the browser source is apparently offline can have really easy reasons up to some tricky issues which we will help you to solve.
Implement the browser source in your streaming software
If the Sound Alerts Extension on Twitch says your browser source is offline, you need to make sure that your browser source of Sound Alerts is active. Make sure you implemented the Sound Alerts browser source correctly in your streaming program. The Sound Alerts browser source is active when you have a scene selected in your streaming program where the Sound Alerts browser source is implemented.
Refresh the cache of the browser source
If the Sound Alerts browser source is not online even if you implemented it in your streaming software and the streaming software is opened, there might be an issue with the source. In that case please try to refresh the cache of the browser source. Open the properties of the Sound Alerts browser source in your streaming software and click the "Refresh cache of current page" button.
Change Browser Source Hardware Acceleration
If this still does not help, go to your OBS Settings -> Advanced and try again with either enabling or disabling the "Browser Source Hardware Acceleration".