that's just Hillary, and maybe leiberman, the party in general is opposed to it, for the most part. when a democrat backs a bill like that there trying to gain votes from people to the right of there party, generaly it's republicans who want cencorship/restrictoins and start these bills.
generaly the democratic party is the party of government helping people, unfortunately helping people costs money and so they are also the party of higher taxes, it also needs more gonvernment infrastructure so they are also the party of big government, generaly speaking. they are secular and intelectual and push for equality (at the cost of personal freedom). democrats couldn't care less about American culture, (the more extreem seeing it as unfair, unethical, greedy, and/or corupt) they are big fans of multiculturalism and diversity and the promotion thereof(equality). democrats also are the party of government distrust, they don't like any secrets, they always suspect the worst in the government.
Republicans by contrast have a bug up there ass about people doing things diferently than they like, they tend to be far more religious and far more trusting of the government (at least while things look good), and have an all incompasing fear of nudeity and foul language, as well as anything that doesn't fit in with a leave it to beaver eppisode (they are 'traditionalists' they like to maintain American tradition and culture (or at least as they see it) at the expense of personal freedom (ironicly)) interestingly they don't have so much a problem with violence (wich democrats sometimes do). now on top of all of this they outwardly have a paradoxical platform of limeted and small government (there monumental failure to deliver this is behind a large amout of the current resentment in the republican constituency). they reconcile the conflict between there traditionalist authoritarianism and the somewhat libertarian position on government by saying that people should have a right to determine how they live on a per comunity bassis, that is if a large part of a town says 'no gays allowed' they have as a comunity the right to determine the standards of behavior. though they will never say it, or probly even think it, republicans are against equality, that is some people will do better in life than others people in power will have prejuduces, and this means that equality is an unatural state, in order to have equality you must take away freedom from people to make bad choices and to punish success and reward failure (as they would put it), to the republican point of view equality(equality of outcome) is unfair. the traditionalist ideology is also the genisis for one of the biggest republican positions militerism, republicans are seen as being a lot stronger on defence than democrats because republicans are seen as not giveing a flying **** about people outside the US, so they are more likely to level another country because it looked at us the wrong way, real threats would be treated even harsher, there is also a strong tradition of militarism in the US, we were born out of a rebelion, we councered the rest of the contenent with individuals with shotguns in wagons, hell we even fought a war with our selves.
a lot of the positions of one party are oftine taken to either be the opposite of the other party for the sake of being contrary or to be more like the other party than the other party is in order to take away a position that makes the other party particularly popular (the democratic push to balence the budget is an example of this, untill the mid 90s the republicans were thought of as the party that ran the government like a bussness and were seen as stronger than the democrats who were thought of as the party of "tax and spend" makeing up huge government projects to solve problems damned the cost)