The Constitution protects your First Amendment right to free speech, but that does not mean you can say whatever you like and get away with it. States have different views on cursing and these views have shaped how they choose to regulate it.
Constitutional interpretation is tricky, especially when it comes to cursing. This is because some choose to categorize cursing under obscene language, an expression that falls under unprotected speech.
In addition to obscene speech, the Supreme Court has ruled that some expressions do not fall under protected speech. These include:
Fighting words: These constitute language that by the very utterance of the person saying it, tends to inflict injury or provoke a breach of the peace. So, if someone came up to you and used profane insults, those insults might constitute fighting words. However, if the person simply carried a banner, or wore a T-shirt saying the same thing, it would be protected speech as there is no face-to-face encounter and no tendency for violence.
It is also important to note that merely using offensive words does not qualify as using fighting words.
Language that incites violence: This is language that is intended to incite people to engage in violence or lawless acts. The Supreme Court in Brandenburg v. Ohio, 395 U.S. 444, 89 S. Ct. 1827, 23 L. Ed. 2d 430  provided further guidance stating speech that advocates illegal activity is protected unless "such advocacy is directed to inciting or producing imminent lawless action and is likely to incite or produce such action."
Although it's probably not a great idea to curse in public, most states won't punish you for it unless it is followed by threats or fighting words. Some states, like Virginia, still have laws predating the Civil War which make "profane swearing" a class 4 misdemeanor. Similarly, in Mississippi, you could be looking at 30 days in the county jail if you profanely swear or curse in front of two persons or more.
You should also note that First Amendment protections do not extend to private organizations. Private entities have the liberty to set their own standards on what is acceptable on their premises.
Even though the constitutionality of these laws can be challenged, few states make it illegal to curse in front of minors. In Georgia, for instance, cursing in the presence of a person under the age of 14 is considered disorderly conduct, while Michigan recently abolished an outdated law that made it illegal to swear in front of a minor.