Who is the man on the cover?

No, the soldier on the cover of the dvd box isn’t Ken Burns’ father – although a photograph of the elder Burns in uniform does appear early on in the first episode, the last in a montage seen within the first 3 to 5 minutes or so of the film. (If you’ve got the dvd set handy, Burns identifies his father during the commentary on Disc 1.) The soldier pictured on the dvd box was ultimately identified (by a family member who recognized him) as part of a graves registration unit. There’s a bit more to that story, the exact details of which elude me at the moment. However, Ken Burns himself related the story during a talk on The War, video of which should still be available via the WGBH Forum Network, at http://forum-network.org/station/wgbh …
(Hunt for Ken Burns The War via the Search box there. The two most relevant results should appear bottom of results pg 1.)

I don’t recall which of those two programs there contains the story of the identification, but it’s one of those two, and both are worth watching anyway.