http://www.thenewamerican.com/index.php/usnews/constitution/1241

In a nutshell, the argument against calling for a constitutional convention is that once convened, such a convention would be free to consider and propose whatever amendments to the Constitution that it deemed beneficial. Which is to say that such a convention could become a “runaway convention” in much the same way that the Constitutional Convention that produced our current U.S. Constitution was a runaway convention that disregarded the guidelines under which it was convened. While most Americans are very thankful for the Constitution produced by our original Constitutional Convention in 1787, most Americans and certainly most state legislators, when fully informed of the downsides involved, oppose the convening of a new constitutional convention in our day.

While it would be perfectly constitutional to convene an Article V constitutional convention, it would not be prudent. Given the present-day general lack of knowledge of both the Constitution and the original intent of our Founders, along with the inordinate influence over our political processes by very biased elites, the chances that a modern-day constitutional convention and subsequent ratification process could change our Constitution for the worse are too great to risk calling for such a convention.

This video explains how our rights and freedoms as secured by the U.S. Constitution are put at risk by state legislatures requesting Congress to call a constitutional convention: