How do I remove Steam Inbox mail?
Solution 1:
If those notifications are comments, you can go to the new comment overview and click "Mark all Read", there is a small link for this at the top, if you have unread comments.
As for inventory and the other categories, you have to go to your inventory and dismiss the new items notification or read any unread chat messages seperately or accept your 300 Gifts. There is no way doing this all at once currently.