this is an excellent point. as it is, the supporters are the poor and uneducated, and the opposition is the middle/upper classes, educated, traveled and more likely to have ties to the US. Chavez would certainly capitalize on visible American support for his opposition.
They would be further emboldened to hear the American president both recognize and reveal to the world, via the world media that can't then ignore it, the facts of their struggle against Cubanization of their beloved country. Furthermore, wide exposure of his support of Iraq, Al Qaeda, Cuba, Libya, Columbian guerillas, bio-weapon lab, and drug trade would make it much harder for the world community to ignore the threat he represents and would invite universal condemnation of his regime.
Yes, Chavez would likely try and rally his core supporters with it, but the UN and OAS and world community would not be able to so readily proclaim his legitimacy or defend him anymore.
Think Iraq, if US govt has any right to complain at all about their WMD's, then surely we can complain about supporters of Iraq and other terrorists, too. Being silent about what's going on in Venezuela, I'm convinced, is not the moral course.
-Shane