Yes, but it is best run by an artistic pastor. Primarily because the goal of an art ministry is to share the gospel with the artists and teach them how to share it with others through their artworks. This goal is often side stepped and forgotten by artists who do not ordinarily practice these ideas in their own work and lifestyle. A liturgical or biblical artist/professor is an excellent alternative to a pastor or priest. But he or she must stick to a curriculum that is agreed upon by the lead minister of the church.
An artist who is not well trained in theology and biblical history may make some serious mistakes about “how” to portray subjects from a Christian world view. Fine arts ministry is a specific field of study with deliberate agendas. It is best run by those individuals who understand the focus of ministry in general as well as have a in depth understanding of the creation of religious subjects. I have met some individuals who are very talented in these sorts of arenas and were not pastors or great artists. But be very careful about employing ordinary artists to undertake fine arts ministry inside of a church. It is not as easy as it may sound.
A music minister sometimes is a very good choice in such cases. He may not be a fine painter, but he could be very talented with communicating art and the gospel in his own field. A fine arts ministry would not be as difficult for him to lead, if he reads some great literature before hand.
Remember that to lead a fine arts ministry one must have fortitude and honesty, but also a great big heart for people. This leader should also be humble so that he does not make decisions that are egocentric in nature. He should desire to teach and lift up others more than himself. Just because art is the vehicle for this kind of ministry, it does not mean that only the best artists should have a voice. That kind of conclusion is a “worldly agenda” and it must be kept within reasonable limits inside of a church art ministry.