When it comes to amending the U.S. Constitution, Republicans have been and remain much bolder. This year's GOP platform (pdf) advocates no less than five specific constitutional amendments: a balanced budget amendment, an anti-marriage amendment, repealing the federal income tax, establishing formal rights for crime victims in prosecutions, and criminalizing abortion. Each of these proposals is a bold, controversial change. By contrast, the draft Democratic Platform (pdf) mentions only one: the Equal Rights Amendment, which today is at once fairly uncontroversial and fairly unlikely to ever move again.
I don't think this difference exists simply because Republicans see more things in the Constitution (and constitutional law) that need to be changes. Many Democratic - or at least Democratic leaders and activists - could probably name several constitutional amendments that they think would better our nation - say, to guarantee a fundamental right to education, or to privacy. Rather, I think the difference represents the parties' perceptions of the political advantages of calling for constitutional amendments. Democrats are far more afraid of suffering political from taking positions that could be perceived as radical or extreme; it will lose them more support than it gains them. By contrast, on many issues Republicans believe that taking a radical stand is a net plus, as there are enough voters and donors for whom strong stands on property rights, federal spending, and/or social issues will win them strong support. It may even be that Republicans today are more used to thinking of bold policy proposals, whereas Democrats are more cautious.