Yes, they are suppressing the vote

For years, researchers warned that laws requiring voters to show certain forms of photo identification at the poll would discriminate against racial minorities and other groups. Now, the first study has been released showing that the proliferation of voter ID laws in recent years has indeed driven down minority voter turnout, and by a significant amount.

In a new paper entitled “Voter Identification Laws and the Suppression of Minority Votes”, researchers at the University of California, San Diego — Zoltan Hajnal, Nazita Lajevardi — and Bucknell University — Lindsay Nielson — used data from the annual Cooperative Congressional Election Study to compare states with strict voter ID laws to those that allow voters without photo ID to cast a ballot. They found a clear and significant dampening effect on minority turnout in strict voter ID states.

“Democratic turnout drops by an estimated 8.8 percentage points in general elections when strict photo identification laws are in place,” compared to just 3.6 percentage points for Republicans.
(Note: ThinkProgress was provided an updated version of the paper [not currently available on the web] that includes data from the 2014 elections. The numbers in this article reflect the updated paper, not the web version linked above.)

For example, the researchers found that in primary elections, “a strict ID law could be expected to depress Latino turnout by 9.3 points, Black turnout by 8.6 points, and Asian American turnout by 12.5 points.”

The impact of strict voter ID was also evident in general elections, where minority turnout plummeted in relation to the white vote. “For Latinos in the general election, the predicted gap more than doubles from 4.9 points in states without strict ID laws to 13.5 points in states with strict photo ID laws,” the study found. That gap increased by 2.2 points for African Americans and by 5 points for Asian Americans. The effect was even more pronounced in primary elections.
The study found that strict voter ID laws had little impact on younger voters as a whole, while there were “small indications” that poorer Americans were adversely impacted, though likely not to the same degree racial minorities were.

Given that minorities tend to vote overwhelmingly Democratic, researchers were left with little doubt that strict voter ID laws were hurting Democratic candidates.