The military spends more on marching bands. To eliminate the NEA would save each American exactly 65 cents a year - the cost of a can of soda. The bigger, more established concerns would take a hit, but would probably survive. Some of the smaller organizations might have to close. But to consider the arts in such a framework is an unfair oversimplification.
Government subsidies are an integral part of many industries, from farming to sports franchises.
One of the better reasons for such subsidies is that they shield enterprises from momentary reverses. If farmers couldn't survive a single bad season, it would ultimately make for a weak social fabric. With the arts, the time frame is often more stretched out. It can easily take decades for general taste to approve of development in art. The last hundred years are full of examples.