Civil liberties are the subject of a great deal of partisanship even though both parties do espouse freedom. The reason for this is that the two parties think of civil liberties and freedoms in different ways.
For Democrats, civil liberties tend to center around personal freedoms to behave as we like in our private lives. This is why Democrats tend to emphasize things like gay rights and abortion rights, which they see as private issues. Democrats also tend to be suspicious of government when it comes to enforcing law and order. They feel that there must be strong protections against things like unreasonable searches and seizures.
For Republicans, freedom is seen in a different way. They are suspicious of government, but they feel that what is dangerous is government regulations of economic freedom. They sometimes feel that the government wants to impose an authoritarian system with the government telling us what we can eat and how we have to keep ourselves safe. Republicans also see government encroaching on traditional freedoms such as the freedom to profess one’s religion in public. They feel that laws against things like prayer in school constitute assaults on our freedoms.
Both sides, then, want civil liberties and freedom, but they conceive of these things in different ways. This makes civil liberties a partisan issue.