How has this not been done yet? *cough* CORRUPTION!*cough*
Pretty simple. The government gives everyone a checking and savings account at birth. No overdraft or anything wild, no credit, just a checking and a savings account, along with a debt card for transactions. Savings accrues interest at federal funds rate, or you could bump it up for certain groups like the young to encourage saving, and obviously cap it for the little richards who will try to abuse it. Hell, you could just administer most welfare though the account.
I'm sure reptiles will go bonkers but basic banking services really are a necessary utility in the modern world. And it's not like you have to use the account if you are worried about ur guns n freedums. Of course, this would suck deposits out of the current banking industry and the rentiers would have to find another basic service to monopolise and overcharge for. Would provide some really interesting data too about actual purchasing power of some of the poorer communities in the country, and even longitudinal studies about wealth and class mobility. This is the sort of "big idea" we would have done on 2 weeks notice back when we were great, when we built highways and went to the moon.