This is a big field, including music, dance, song, painting, poetry, literature, plays and film, as well as combinations of them and possibly newer types of creative communication. Language, with all the implications sketched out above, is involved in song, poetry, plays and film. The other arts are more or less language-independent.
Language-dependent or otherwise, most art forms are quite tied to the nation state at present. There is not much danger that an Impressionist picture of Paris will be mistaken for a Social Realist townscape in Siberia, or a Mexican bodega painted by Diego Rivera. But that is perhaps mostly because the subject matter is national in character. Stylistically, impressionism was more or less the same everywhere, with some time lags, as in music, the forms of musical composition tended to vary just about in step across a wide swathe of countries (cultures). There is nothing 'national' about the string quartet, even if it did originate in Austro-Hungary. Composers from all countries have used that form, and still do, although it is a bit out of fashion at the moment among composers. The same goes for popular culture and fashion. Styles have always crossed national boundaries. Indeed, that is just one more proof that it is language, and the nation states which made it into their competitive banner, which is to blame for the antagonism of different linguistically-based cultures.
It is significant that the best writers are widely translated – always said to be 'universal' – Shakespeare and Dostoevsky for example. Where is national culture in this?
The arts are perhaps more a reflection of cultural norms than a cause of them. If national languages remain, they will be expressed through the arts; if a common language develops, likewise. Images are less tied to nation states; or at any rate they can be understood widely even if they are national in character. An image of the destruction of the World Trade Center in New York in 2001 may cause different reactions in different people; but its objective reality is crystal clear to almost everyone on the planet. Everyone who watches television, that is.
In so far as the arts reflect the reality of the world as it is, they have already become globalized, through images such as those of the twin towers, and millions of other, less terrible happenings. That means, surely, that the image library in people's heads has already become globalized? A high proportion of people from almost all nations and cultures has seen the Berlin Wall being knocked down, the Titanic sinking (real or imagined, it doesn't matter), a man walking on the moon, the Great Wall of China, a man and a woman riding horses on the beach, the tsunami, cherry trees in Washington, a Russian spy dying of radiation poisoning, baby seals, the World Cup (there is only one!). This list of universal images could be 1,000 times as long, and you would still have barely started cataloguing the shared picture library of human beings. It is already globalized, and will not be turned back.
The existence of a common (globalized) library of images, cognitive concepts and cultural 'memes' implies the possibility, as mentioned above under The Media, that small niche groups can form on a globalized basis. That has always been possible on a national or local scale. Although held back by communication limitations, there are of course such groups in existence as the The Friends of Battle Arts Festival, the Rome Operatic and Dramatic Society, Nevada, and the Omsk Association of Balalaika Players. There would seem to be no limits, through the Internet, of the scope for formation of a bewildering number of affiliations of this type, especially when language ceases to be a barrier.