The use of SSNs began in 1936, as a result of the Social Security Act of 1935 and were originally intended just to be for the social security program. In the 1940s, the numbers were used to help identify citizens in new record keeping systems. It wasn't until the 1960s-1970s that SSNs really began to be used to verify identity for taxes.

Let's think about taxes for a moment. The current system of today is pretty sophisticated and far reaching. Many taxes as we know them, federal income tax for one, are less than 100 (even 50 in some cases) years old. Taxes on alcohol, tea, etc would be taken at time of purchase.

Let's consider the first federal income tax. In the early days, however, it would be up to towns/cities to personally collect taxes and record whether an individual paid or not. As census statistics were taken beginning in 1790, governments could track the number of citizens in their location. But, pre-WWI, there was really little need for identification cards (though they would get more in style in the early 20th century with the improvement of photo technology). To apply for credit or insurance pre-camera days, a signature would be required on a contract. Could you skip town? Of course, but the local police will be notified and you will get a warrant out for your arrest. Scam one to many people and there will be consequences.

Source: Behind the numbers (history of Social Security and Medicare) by Alan Brinkley (p Bait and switch on Social Security: it wasn't created as 'insurance,' but now everybody thinks it is by Phillip Longman (printed out from a previous course of mine, can't find links) Wikipedia (for the dates for the Census and info on the camera ids)