Not very much, and maybe not at all. There are numerous papers on this question, with a variety of results, but this one is perhaps the most thorough and best-specified:
Got ID? The Zero Effects of Voter ID Laws on County-Level Turnout, Vote Shares, and Uncounted Ballots, 1992-2014
Abstract: Abstract Do voter ID laws disenfranchise voters? Despite a ferocious, politically charged debate in the media and public opinion, the scholarly evidence on the effects of voter ID requirements is inconclusive. In this paper, I use county-level administrative data from 1992 to 2014 and a Differences-in-Differences research design to identify and estimate the impact of voter ID laws on turnout, Democratic vote share, and irregular ballots. I find no effect of ID laws on any of these outcomes. All estimates are fairly precise and robust to a number of regression specifications. Estimates of heterogeneous effects by educational attainment, poverty rate and minority presence are similarly supportive of ID laws having no impact on electoral outcomes of any type.