The Church of the Transfiguration, an Episcopal congregation, became connected with theatre when a neighboring church refused to perform a funeral for an actor. The snubbing priest mention that "there's a little church around the corner where they do that sort of thing," and The Church of the Transfiguration has been called "the little church around the corner" and served actors ever since. Today, the church's main support has been institutionalized and separated from the official working of the congregation. It now takes the form of the Episcopal Actors' Guild, which provides "[c]haritable help for performers of all faiths, and none." The Actor's Guild provides direct relief to performers in the form of money for rent and medical bills as well as scholarships, network connections, and a variety of other services. At this point it's only loosely affiliated with the Episcopal Church, it's main connections being a few priests on its board, being housed in the Little Church, and an awareness of the religious communities of all sorts that support the arts. The office in Manhattan is its only branch, but Executive Director Karen A. Lehman told me they'd be happy to open another site if anyone wants to set it up.
The Roman Catholic Church of Saint Malachy's, known as the "Actor's Chapel," has historically more actively sought to serve the Broadway community on a spiritual level. Simple acts such as having services at midnight and 4am in order to accommodate the schedules of theatre workers, and encouraging performers to come in, say a prayer and light a candle before opening nights, have gone a long way in making this church the haunt of string of stars. It also didn't hurt that a long line of priests have made it a point to get to know the artists in the neighborhood, visiting them in green rooms and dressing room, and providing comfort to actors' unique fears of failure and age, and the public venue in which they must live their lives.
Both the Little Church and the Actors' Chapel seem to have more of a historical connection to Broadway at this point. However, Middle Collegiate Church, located in the East Village of Manhattan is actively cultivating its community of artists. Middle supplements its worship with two sets of artists in residence, an organist, and the Omega Dance Company, as well as with a gospel choir. They also actively encourage the artists in their congregation to organize events in which to share their talents. Literature professors from New York University put on poetry readings by their students and colleagues, and a member's improv troupe performs in the church. Members' talents are identified through a form which they fill out when they join in which they describe their passion and talents; a minister then meets with them one on one to figure out how where the member might best express his or her own sense of vocation, or if the member might want to create something new.
So as not to let my Christian bias completely overlook the ministry of other faiths to Broadway, I should mention The Actors' Temple. It seems to fall largely in the historical theatre connection category, though they've recently started their own theatre company.