The unemployed that would be affected by a move from unemployment benefits as they currently stand to a program like the CCC vote, their families vote.
With this in mind, it is impossible to enact a program like the CCC today.
Nah, it's nowhere near impossible. You could absolutely pitch a CCC program in this modern era if the incentives, pay, social networks, etc. were properly put in place. Some motivating factor, greater good or combination of both would need to be instituted and touted by someone of great influence; say, the POTUS.
To me, the most obvious core component of this program would be some sort of green initiative. Let's assume a modern CCC enrolled only 18-25 year olds the way it did during the New Deal; the reason CCC targeted those age groups was partially because youth unemployment is always higher, partially because they're in better physical shape and partially because training youths presumably has longer-lasting positive economic effects. From that green perspective, it's not hard to imagine an environmentally friendly POTUS who's got a good rapport with universities, teachers and unions, pitching the idea of youth labor, both physical and skilled, to help to beautify and improve communities. Some specific green initiatives could be something like Habitat for Humanity style home building and rejuvenation for neighborhoods whose states have prioritized building elsewhere. It could be clean air and water initiatives in lakes, ponds and swamps. It could be social media outreach training and organizing, with labor unions obviously being key here. These are things youths are already far more interested in and attuned to than the youth of the 1930's. Not that that generation hated the environment, I'm sure they appreciated it and I'm certainly not an expert on the social construct of that decade, but I'd bet dollar to donuts they were simply nowhere near as informed and/or interested in the environment as our young whippersnappers are today.
I'd be pretty excited to be a part of a modern CCC program if I were an 18-25 year old, but there are issues. You would need as I said the right incentives and would be absolutely have to be paid for responsibly. To pay for it, you'd have issues of whether CCC would be voluntary, but if you made it mandatory with a small fee exemption (like the mandate), it'd work. Charge a small fee that's just large enough to incentivize enough youths to enroll to avoid said fee, but small enough that enough will just pay the fee to avoid CCC, which both helps raise money for this project (which obviously desperately does need to be paid for, lots of administrative overhead in this program) while also enrolling enough youths to make it work in the first place. Some incentives can also include school credit, good pay of some sort, internship-style stuff but without exploiting their labor at depressed wages the way some corporations take advantage of unpaid interns. CCC I think might inevitably result in some trade union opposition due to the low wage issue (though perhaps not enough to matter) and perhaps most significantly you'd see pushback from over-25 laborers who will be crowded out from these economic opportunities due to their age; construction workers if we're doing Habitat for Humanity stuff, undocumented immigrants, etc.
In any case, a marriage of high schools, universities and various local or national businesses working in concert to train and employ youths that helps build infrastructure frankly sounds great to me personally.