What would be the easiest way to box out the I beam in my basement
And loose the least height? I would like to have a wood finish.

Hi
To box a beam, cut a 1X board the width of the beam, Cut it a tiny bit (1/16") so you will gave a tight joint later.

Mount it to the bottom of your beam, If it is steel you will have to mount it with construction adhesive.

Now measure from the ceiling down to the bottom of the board you just mounted and ADD 1/4" to your measurement. This extra 1/4" will hang below your beam bottom. It looks better than trying to match the corners. The only thing that may look better is to miter the corners and match them at the corners, but you need to be a pro woodworker to do this.

Now mount the sides of the beam, nailing into the bottom board you mounted on your beam, with finish nails.

In shorter terms you are wrapping your beam on 3 sides with wood and will only lower your height by 1".

Good luck Roger