Vouchers take funds away from an already impoverished public school system. This money should be spent to improve public education. Many private schools are religious schools, and it would be unconstitutional for public money to be spent to support these institutions. Many parents would still not be able to afford to send their students to private schools. These students would in effect be left behind.
Public education is a failure.
Throwing good money after bad will not improve a hopeless situation. The unifying influence of public schools is non-existent. The wealthy have already removed their children from public schools, or they have fled to suburban communities. The children who remain are left in a state of defacto segregation. The voucher is given to the parent to spend as he or she sees fit.