There are many reasons for asking this question. Politicians, educators, news reporters and cultural affairs directors all have reason to ask it. All must make decisions about how to regard, fund and support "the arts"; all sometimes find it confusing to make decisions about movies, paintings, jazz, rock, rap and classical music performances, dance, theater, pottery, and perhaps even stand-up comedy, juggling and "the culinary arts" as if they all belonged to the same category.
Artists also have reason to ask what the arts have in common. A songwriter asks it, at least implicitly, whenever she or he tries to match music and lyrics. If music and poetry don't have anything in common, it shouldn't matter what music you set the words to; but it obviously does matter. And it matters in a number of ways. Both music and poetry have meter, and both have regular or irregular phrases. So the meter and phrasing of the music should match the meter and phrasing of the lyrics. But so should the mood of the music and the lyrics match. A bouncy tune will clash with serious lyrics; a sad or grand tune will not fit with silly lyrics, except as a joke.
These facts seem so obvious as not to need stating. Yet they raise important questions about music and poetry. After all, these are very different art forms. Poems, being language, have meaning as a matter of course. But music does not "mean" in the same way that poetry does. So how can a piece of music fit some meanings better than others? Asking this question immediately opens up a range of intriguing philosophical questions about the nature of musical meaning. There has probably been more disagreement about these questions among philosophers (and musicians and critics) than about any other single question in the philosophy of music. (See the page on Suzanne Langer for more about this question.) At the philosophical level the questions remain difficult. At the practical level they are answered every day by songwriters and composers.
Similar questions lie behind the practice of the film-makers art. Film includes not only moving images, but also sound. It is like music, even when it is silent; and it uses music. It is like theater; it is like painting; it is a form of photography. How do these different elements work together in a movie?
So what do the arts have in common? As a discussion starter, what about this (most of the credit for this idea goes to the 20th century philosopher Ludwig Wittgenstein):
The arts are like a large extended family. There are many family resemblances among them. Some recur frequently; others are shared by only a few members of the family, or are unique to one or two members. There is no one defining set of characteristics such that all and only "arts" have those characteristics. So an attempt to define "art" is bound to fail. Questions about what particular art forms have in common, on the other hand, are frequently of great interest, and the attempt to answer them may be both illuminating and as stimulating to the practice of those arts as to aesthetic theory.