Bandana was my first thought, but one feels the need to clarify that the bandana is over the person's face, as bandanas are also worn around the neck and on the head.
To signify that it's covering the face, the best single word is mask.
From Wiktionary (emphasis added):
A cover, or partial cover, for the face, used for disguise or protection.
So masks don't need to cover the whole face, they can be a partial cover.
To tell someone to put this mask on may depend on where it is to begin with. Often times a bandana will be worn around the neck, and then raised to become a mask, which would make "up" make at least as much sense as "on". You might say "masks up!" or "masks on!". Less tersely, you might say "put your masks on" or "put your masks up". More tersely, a robber may simply tell his crew, "masks!".
Really though, the best way to refer to this item is as a "bandana mask". It's not one word, but it clearly indicates both the material and the location. Go to Google, and do an image search for 'bandana'. Very few, if any, of the results will be over someone's mouth. Now search for 'mask'. Very few, if any, of the results are bandanas. Now, search for 'bandana mask'. You get exactly what you're looking for.