So a boxer and a grappler step into the cage...
The reality is that, in a one-on-one situation, ground-fighting beats stand-up striking up to 90% of the time. Take a stand-up striker, dump him on his back via a single- or double-leg takedown, and start working through his guard. Or, better yet, let him make the elementary mistake of turning his back to you when he's trying to get back up.
And that's it- you've got him. He's done. His ability to fight when he's on the ground is completely removed, because you've nullified the advantages of his biggest and most powerful weapons and can now work him over more or less at your leisure.