In the '60s, most Americans who got SS cards did so for one of two reasons -- they were entering the work force or taking the SAT test.
Not being an early member of the work force, I had no SSN in 1968 and got one so I could take the SAT.
It was a relatively simple process. My mom drove me down to Richards Street (?) in downtown Honolulu to the SS office. I had my birth certificate with me. The birth certificate was to establish birth date and, hence, age. Identity was not an issue. The process resembles the application for a Taxpayer ID Number (TIN) for a business, trust, or estate today. The number was merely designed to distinguish one taxpayer from another.
The SS office was right off the street with no security of course. No security in those days. I walked in, filled out a short form at a stand-up table. This was the same basic form used since the program's origins in 1936: Name, Address, Employer, DOB, Place of Birth, Parent's Names.
I handed my SS-5 form and birth certificate to a clerk seated at a desk with a typewriter -- minimal or no barrier. She rolled a strip of SS cards into her typewriter and typed my name in the blank, tore it off, and handed it to me for my signature. Although I don't remember, she would also have used a mechanical number stamp to put the number on my SS-5 form to make sure that it was connected to the right number.
Like all SS cards of that era, it read: For Social Security and Tax Purposes--Not for Identification.