Both Democrats and Republicans claim to value the liberty of the individual. Where they differ, and they differ quite strongly, is in their approach to protecting it. Democrats tend to advocate the power of government in this regard, seeing government as the only institution capable of ensuring that individual liberty is properly protected. They look upon liberty in a much more positive sense than Republicans, emphasizing the freedom to do rather than freedom from.
In that sense, Democrats look upon government as an enabler, giving people liberties that might otherwise be curtailed by poverty, bad education, lack of opportunity, and so forth. To that end, they argue for a much greater role for the government in the running of the economy. In this reading, governments shouldn't just set the rules, they should regularly intervene to ensure that the liberty of the individual is maximized through increased educational opportunities, better healthcare, and improved wages and conditions. Only in this way can individual liberty in its fullest sense become a reality.
Republicans, on the other hand, tend to believe in negative liberty, or freedom from. They see government as being the main threat to the liberty of the individual, a mighty colossus that is always looking for ways to increase its power. To that end, governments should be restricted in the powers that they have, especially in relation to economic affairs. The more limited government is, the freer the individual.
That's not to say that government doesn't have a role to play in protecting individual liberty. Republicans freely acknowledge that government must set the rules to ensure that the individual's rights and liberties are protected by law. But beyond that, the government should not intervene. For Republicans, when it comes to individual liberty, the government should be concerned with its formal requirements, not with substantive outcomes. In other words, it's not for the government to impose its vision of what a free individual looks like.