Support is strong when you ask a one-dimensional question ("do you, in general, support voter ID?").

The poll linked from the CNN article demonstrates this: there's also overwhelming support for making voting easier, but voter IDs (without free and accessible ID services) will make it harder. And that's the crux of the common Democratic position: voter IDs are perfectly fine if we ensure access to IDs. But that needs to happen before or with any restrictions on voting, to prevent disenfranchisement.