It would cost the public schools because state and federal program reimbursement is usually done via student census.  Fewer students means fewer dollars in the budget.  Private schools are free to take whomever they want and refuse whomever the want.  As Mr. Crum pointed out, the wealthy can pay the balance of tuition costs, leaving the poor, the problematic (read un-parented), and the special needs students behind in the public schools with fewer resources to serve them.

"In a voucher scenario, EDUCATION would improve..."   yes, for the wealthy families that could afford the balance of the tuition costs.  

The educational voucher system is just another form of a "gated community."  Keep out the riff-raff (poor).