State and Federal employment records already exist with your social security number, address, employer address, employer ID number, and amount of taxes that have been withheld. I can retrieve my own copy of this record online from the IRS and my own state relatively easily.
Someone should not be able to [automatically] claim unemployment benefits for past income on which taxes have not been withheld, or estimated taxes have not been paid. That seems trivially fair. Even people making minimum wage will have payroll taxes withheld. Calculating an upper limit on the amount that can be processed automatically should therefore be trivial.
Next is the question of where to send the payment. DMVs cover a large percentage of the population. Beyond the DMV, there's voter registration, credit reports, address of record for where the last tax refund was sent, utility bills, phone records.
When I had a larger than typical refund last year, my state sent me a letter asking for copies of my and my wife's social security cards, passports, and utility bills before they would issue the refund. I felt that was pretty burdensome considering I was requesting the refund to the same exact bank account number which I had been paying them from for the last 5 years.
It's obvious the state is just totally incapable of running even the simplest of validation rules. It's just helicopter money dropping from the sky, and they don't really care who gets it, because it's not their money. It's not like anyone is going to lose their job or not get re-elected because of it.
The fact that's it's so easy to funnel all this money out of the country is a pretty wild indictment of FinCEN and the like.