PowerBlog rightly notes the dangers of funding faith-based initiatives to the charity itself because the tendency is to become increasingly dependent on the large contributions of any benefactor, in this case the government, and to become correspondingly beholden to the interests of that benefactor. The flip side is that faith-based groups are effective where government is not, e.g., substance abuse, because they are based on embracing the dignity of each human life.
The autonomy question is real, but can be mitigated by using approaches common in other areas of conflict of interest such as a merit-based grant program administered by an independent body. Another possibility would be expanding the tax breaks of small charitable donations to broaden the funding base from individuals and corporations who would be given an incentive to make many small donations to many organizations and diffuse their influence correspondingly.
If government is going to be in the business of providing welfare, which does not appear likely to change soon, then we ought to be driving efficient and effective use of our tax dollars. That is why, despite the potential downside, we ought to move toward funding faith-based initiatives.