Beetles aren’t kosher.
Carmine is another name for the cochineal beetle.
Here’s a list of what is and isn’t kosher. It’s just a general guide because not all rabbis agree on what is and isn’t kosher which results in there being about 360+ kosher certifying agencies around the world. Consult your local Orthodox rabbi for more specific information.
http://www.star-k.org/kashrus/kk-palate-secretingredient.htm
That sure sounds like an odd thing for Starbucks to want to use, as it would shut out religious Jews from their clientele.