Regarding cost, it should be paid by the state. At least it's so in Mexico
Someone who can't take time off work to get an ID most certainly can't take time off to go vote.
In that case you can still use it to work, buy alcohol, drive, get government benefits, etc., but you still might need to spend four hours at the DMV in order to use it to vote, because your ballot is customized to your particular city/neighborhood.
So, it's still a tax/restriction which can prevent voter participation.
https://www.politifact.com/factchecks/2012/jul/11/eric-holde...
https://www.aclu.org/documents/oppose-voter-id-legislation-f...
https://www.usccr.gov/files/pubs/2018/Minority_Voting_Access...
https://www.washingtonpost.com/politics/courts_law/getting-a...
https://www.vox.com/xpress/2014/11/4/7157037/us-voter-id-req...
https://www.npr.org/2018/09/07/644648955/for-older-voters-ge...
https://rewirenewsgroup.com/2014/10/16/well-actually-pretty-...
https://www.theregreview.org/2019/01/08/shapiro-moran-burden...
https://www.theatlantic.com/politics/archive/2014/10/heres-h...
https://scholars.org/contribution/high-cost-free-photo-voter...
https://now.tufts.edu/2018/01/23/proving-voter-id-laws-discr...
Why is it not racist or discriminatory against poor people in the rest of the world?