I'm a fan of actual evidence, and Lotstodo brought actual evidence to the table. But, it is worth noting that even if you take the Social Security shenanigans off the table, under the Clinton administration, the federal deficit came within $18 billion of being eliminated. (deficit, not debt) Whether even this tiny victory would have happened without the Republican congress is open to question, but history does not bear out that Republicans are better at fiscal management.
https://www.cbo.gov/publication/41880?index=10871
Nevertheless, my main point stands: Republicans don't have a good record on fiscal restraint. Bush and Reagan couldn't manage it even with Republican control of the Senate (in Reagan's case), and Republican control of both houses (in Bush's case).
I actually assumed this happened because of the wars Bush was fighting. According to the US Department of the Treasury, though they were a big part of it, they were not the biggest part of it. This 2011 chart also includes the beginnings of the Obama Administration. (As a parenthetical, it also shows that CBO projections aren't that great.)
It is now 2015. Here is a simplified but telling article that compares recent presidents. Every one of them grew the debt. It also points out that, as GuardDad said, Clinton benefitted from a booming economy. But it also tells me that I shouldn't associate either party with fiscal restraint.
http://www.usnews.com/news/blogs/rick-newman/2013/02/20/why-a-bill-clinton-style-budget-miracle-wont-happen-again
I'm not going to take a candidate seriously who says they will reduce the debt or the federal deficit until they put all the sacred cows on the table with specifics: military spending, Social Security, and raising taxes. It's fun talking about cutting spending or sacrifices in general, but people don't like it when politicians get specific.