Speaking nationally, Democrats currently have a 6%-7% advantage over Republicans in terms of voter representation. In fact, Democrats have a lead among every demographic group except white males without a college education. However, Democratic voters tend to be clustered in urban and suburban areas, which attenuates their voting power somewhat and leaves them vulnerable to manipulations like gerrymandering. Further, they have lower SES as a group, meaning they are more affected by restrictions on registration and voting. And that's not mentioning the Electoral College, which gives a significant advantage to parties that control smaller states (which currently — for the most part — lean Republican).
So yes: making voting more restrictive mainly impacts low income voters in urban areas, those who have the least free time and fewest material and financial resources to jump through bureaucratic hoops.
At least some Republicans and conservatives explicitly advocate for this kind of implicit disenfranchisement in order to maintain power. I cannot speak to the intentions of the GOP as a whole, except to note that anyone in the political universe with any minimal competence is aware of the fact that there are material conditions to voting which can present obstacles. This has been well-known and thoroughly argued since the Jim Crow days, back when the Supreme Court struck down poll taxes, so anyone who talks about voting restrictions without also considering the issue of structural disenfranchisement is either deeply ignorant or purely Machiavellian. Granting that some conservatives do argue that there should be a 'civic commitment' standard applied to voting — akin to the ancient Greek practice of restricting the democratic participation to established property owners and native sons, on the grounds that such people have a firm commitment to the welfare of the community — the GOP itself has never raised that explicitly as a platform, for the obvious 'optics' reasons.
And note that the situation is more complex than it appears on the surface. The GOP could (for instance) work to increase its voter base, thus obviating the need for voting restrictions. But that would involve creating a forward-thinking platform meant to appeal to a broader coalition, which risks insulting certain die-hard, single-issue voting blocks that the GOP has catered to since the 70s. Those voting blocks are not exactly unified, but they represent some of the most readily mobilized elements of the GOP base, so the GOP prefers a fragmented, decentered, 'talking point' style of politics, one which allows them to play both ends against the middle without committing themselves to anything. Squeezing out unwanted voters is simpler and safer.