Voucher/tax credits starve the public school system of resources/funding.
It depends on how it's structured. Most voucher programs only send the variable costs to the private school. In other words, on a per-pupil basis, the public schools usually come out money ahead. However, that doesn't satisfy the voucher opposition. They hate it anyway.
Charter Schools have mixed success.
In areas where we have charter schools (which aren't the same as a voucher program), the public schools usually don't have mixed success. They're usually terrible. Very rarely is a kid worse off in a charter school.
Of course, choosing your students is something the public schools don't have the benefit of.
You don't have to give participating private schools that option either.
Just to be clear, I'm not a supporter of a national voucher program. That should be a state's prerogative. However, states with poor performing public schools should give kids a choice. They shouldn't be screwed.