I would like to approach profanity from two standpoints: Jewish law and American jurisprudence. I want to look at the issue Jewishly because 1) I'm Jewish, and 2) Judaism influenced Christianity, and as such, has impacted this bi-polarity. I also want to bring American legal arguments because it's interesting to see how "life, liberty, and pursuit of happiness" interact and/or conflict with Jewish values.
Profanity in Jewish Law
Jewish speech ethics are not only nuanced, but they also set a high standard. Since G-d created the universe by speaking (read Genesis 1), Judaism teaches that words have a powerful impact. The Chofetz Chayim points out that speaking לשון הרע (literally "evil tongue," denotes "bad speech ethics"), one can violate up to 31 mitzvot. In Jewish law, it goes well beyond avoiding slander and libel. Gossip, slurs, and negative truths are forbidden under Jewish law. If such restrictions are in Jewish speech ethics, it would be safe to assume that the high bar set also applies to swearing, and you would be correct. Foul language (נבול פה) is also forbidden under Jewish law.
One of the main reasons not to swear under Jewish law it degrades another individual, which is prohibited because we are all "created in His image." But what if you are using swear words to convey intensity of a given feeling? Some people might. Most don't. Since swear words are used so frequently these days, they have lost their intensity, but have not lost their meaning. Plus, if an individual has a sufficiently large vocabulary, he'll find a way to express the same intensity without resorting to swearing.
If you think about it, most swear words are focused on either sex or functions performed in the bathroom. If Judaism teaches that we should elevate the mundane to a level of holiness, especially speech, then swearing is not the way to go about it. Rabbi Elazar ben Yaakov (Derech Eretz Rabbah 3:3) compares swearing to pollution: "A person who uses rough language is like a pipe spewing foul odors in a beautiful room."
Given the amount of swearing in our culture, the only way to have a completely swearing-free zone is to live in an insular community with those who have similar values to you. For most of us, however, we hear swear words often enough. For those of us who want to cut back on swearing, we try to apply such high standards to our lives, easily forgetting that we are human, and thus prone to err. Although we might swear from time to time, we have to accept our inconsistencies if we are to minimize the gap between theory and practice in our daily lives.
Profanity in American Law
It would be nice to close the gap between in theory and practice not just in my private life, but in greater society. If there were a society free of swearing and people speaking unkindly of one another, it would be spiritually liberating, akin to not having to inhale second-hand smoke all the time. It might explain why nearly two-thirds of Americans favor FCC regulation of profanity.
As nice as it might sound to have the government regulate profanity in the name of moral decency, there's this little thing called the First Amendment. Aside from exigent circumstances such as one's speech inciting violence or causing harm to another, the First Amendment gives an individual the right to free speech. That is why the Supreme Court has upheld such acts as desecration of the American flag (Texas v. Johnson), a Nazi rally in a predominantly Jewish neighborhood (National Socialist Party of America v. Village of Skokie), owning pornography (Stanley v. Georgia), depictions of animal cruelty (United States v. Stevens), and the right for the Westboro Baptist Church to disseminate its hate message on public property (Snyder v. Phelps).
There is no constitutional right to not be offended. As repulsive as these acts might be for a good majority of Americans, they are protected by the First Amendment, whether we like it or not. As tempting as it might be to censor things such as swearing, if we do, it's only a hop, skip, and jump away from having censorship bureaus and have the level of censorship they have in China.
Additional Thoughts
If you're worried about your children swearing, as I mentioned earlier, you can live in an insular, religious community, such as the Amish or the the Satmar Chasidic community of Kiryas Joel. If that doesn't work, censor television programs that have swears in them. If that doesn't work, have a mature, one-on-one discussion with your child about swearing and how to handle it. Either way, it's called good parenting.
And for those of us that are adults, we are grown up enough to make our own decisions about what we view and what we hear. In America, if we cherish "life, liberty, and pursuit of happiness," we should be able to choose the level of obscenity of our media. If you don't want to deal with swearing, shut off the television or find more suitable programming.
Why should moral puritans and those of the Religious Right, who are a minority, get to dictate the programming of American citizens? In a free society, one of the joys I get is free will. Free will gives us the ability to make moral decisions. I should be given the choice of whether to listen to or use profanity. If that free will is taken away from me, I don't have the ability to actualize my potential in the moralistic sphere.
To summarize my Jewish libertarian standpoint, it would ideally be nice not to have to deal with profanity. It is spiritually healthier for people. However, we don't live in that ideal world. We have to deal with the pervasiveness of swearing in our culture. As much as a lot of us do not like profanity, there is that reality that we ourselves will give into it because we are human. But we should be given the latitude to deal with it. Those who crusade against profanity shouldn't erode the Constitution to bring about their misconceived notion of utopia. Rather than treat us like children who cannot handle the concept of profanity, maybe citizens should be treated like adults and figure out how to deal with profanity on their own terms.