When exactly does a church's board of elders (Baptist) step in in this situation?

It depends entirely on the depravity of the sin and the position of the sinner in the church. Standard approach is to counsel first one on one and in private. Next is by two elders, next is by the full board. If the depraved is a board member or deacon it would go before the full board at the first step because no error is allowed in the representation of the church. If the sin is so grievous that it puts anyone else at risk as well it would go before the full board. Pedophilia for example or theft from the church.