This will do absolutely nothing to prevent those actors from accessing Claude... they already recruit young unemployed Americans to do proxy job interviews[0][1], etc. They'll just pay young unemployed Americans to do verification for them.
That sounds likely to increase their costs and create new opportunities to get caught. Not a silver bullet but not "absolutely nothing". Like how anti-money laundering laws don't wipe out all crime, but are still worthwhile.
If the API costs are gonna be thousands, or the subscription will be $20/month, is it really that expensive to pay some guy on Discord a $50 gift card to verify the account as a one-time setup? Better yet, we'll probably start seeing fake porn websites and other phishing sites that ask to verify your age but end up proxy verifying a bunch of these services in an automated manner with minimal costs, and you'll be able to buy verified Claude accounts for tens of cents on account marketplaces. Just as you have been able to buy verified Discord accounts, aged Steam accounts, etc...
[0] https://www.tradingview.com/news/cointelegraph:6192f38e3094b...
[1] https://youtube.com/watch?v=QebpXFM1ha0