It seems to depend on who you ask, and they will depend on different interpretations of the same data gathered since HAVA went into effect.
If you ask John Lott or the Federalist Society, voter fraud is a known problem, with increasing persecutions and dire consequences, while increasing voter ID requirements actually seems to produce more voters (and campaign finance laws reduce voter turnout). From John Lott:
The results provide some evidence of vote fraud and that regulations that prevent fraud
can actually increase the voter participation rate. It is hard to see any evidence that voting
regulations differentially harm either minorities, the elderly, or the poor. While this study
examines a broad range of voting regulations, it is still too early to evaluate any possible
impact of mandatory photo IDs on U.S. elections. What can be said is that the non-photo
ID regulations that are already in place have not had the negative impacts that opponents
predicted. The evidence provided here also found that campaign finance regulations
generally reduced voter turnout.
So, according to this one (conservative) commentator, non-photo ID laws do not affect turnout, while the jury is still out on whether or not photo ID laws do affect turnout.
Lott's arguments seem to mimic those of @user1873; namely, that participation increased, so therefore there could not have been a suppression of the vote. Later in the paper, he states:
How did these laws impacted voter participation rates? As a first crude measure, I only
considered states that had changed their laws over time to compare how the participation
rates changed when the laws changed. Obviously this simple comparison ignores that
many other factors are changing, but it at least compares only the same states over time.
He goes on to provide a number of other metrics, but they all seem to rely on the assumption that the introduction of a voter ID law should result in a lower number of participating voters. To me, this seems like a shaky assumption, when such a small percentage of potentially eligible voters is participating in any given election. If only 50% (or fewer) of people are voting, then increasing to 54% does not mean that some people were disenfranchised, it just means that more people decided to go vote. People could still have been disenfranchised, and not reflected in final vote tallies.
Richard Atkinson argued in 2007 that the logic in Lott's article is incorrect:
This paper finds that photo ID requirements fail to fulfill their primary purpose (the prevention of fraud); in fact, photo ID requirements decrease legitimate voter turnout (and therefore may increase the impact of fraud)
Unfortunately, it's paywalled, like many other legal documents on the subject, which limits my ability to present the results here.
Meanwhile, Alvarez et al noted similar effects, going further into the data than Lott:
Looking first at trends in the aggregate data, we find no evidence that voter identification requirements reduce participation. Using individual-level data from the Current Population Survey across these elections, however, we find that the strictest forms of voter identification requirements - combination requirements of presenting an identification card and positively matching one's signature with a signature either on file or on the identification card, as well as requirements to show picture identification - have a negative impact on the participation of registered voters relative to the weakest requirement, stating one's name. We also find evidence that the stricter voter identification requirements depress turnout to a greater extent for less educated and lower income populations, for both minorities and non-minorities.
Figure 5 of the paper shows:
Given that this second paper acknowledges the incompleteness of the aggregate data in describing voter behavior, and then proceeds to address that lack in a closer examination of individual behavior, I believe they have produced a more robust argument that stringent voter ID laws lead to voter suppression.