Can stalwhert goatherd kill blade of the exile?

Yes. Blade of the Exile has an Energy cost of 3, which is within Stalwart Goatherd's "kill a gear with Energy cost 4 or less" trigger.

Being attached to a unit doesn't protect it — attached gear is still an independent Gear on the board and a legal target for anything that says "gear" (Rule 148.1.a.3). If you kill it, the Blade detaches and moves to the trash, and the unit it was attached to just loses its +3 Might and Ganking; the unit itself survives.

Note the trigger is a "may," and it happens when you play the Goatherd — so the Gear needs to already be on the board (at Base or attached at a battlefield) to be chosen.