I have been playing some percussion acoustic guitar like the song http://tabs.ultimate-guitar.com/s/sungha_jung/billie_jean_guitar_pro.htm.

What I am wondering is the proper technique to create percussive mutes that mimic the beats of a drum. Currently I have been using the side of my right hand thumb and slapping the bottom strings, but this of course leads to unintentional harmonics at times, so I have to try mute the strings beforehand. Of course this is not always possible and I frequently end up with ringing open strings, so I question whether I just lack practice or if there is simply a better way.

Is there an easier method for accomplishing drum-like percussive beats without moving too far from the strings (or preferably, staying on the strings so I can quickly pick afterwards)?

There is no set technique to do this. People just seem to do their own thing...and they become pretty damn good at it. I've been looking at lessons by Dale Campbell on YouTube - check him out. Also, Rodrigo y Gabriela have released some lesson-type things, but Gabriela is who you really want to look at for this kind of playing. She's a monster.
And if I'm not mistaken, John Butler too. My one friend plays a bit of his stuff but I forget whether it's more of a tapping style or a percussive style (or both)...anyway, check him out too for what it's worth
I was actually rewatching one of Dale Cambell's video as you posted this :p I will be sure to check out this Gabriela guitarist as you suggested. Thanks!