First you need to understand ther context of the situation. The Taxi industry in the 1930's was not nearly as expansive and large scale as it was now, and not nearly as regulated. The 1930's was an iconic time, because, if you recall, this was the era of the Great Depression. Jobs and work were hard to find. Money was scarce.

However, if you had a car, there was a single job you could do with ease.

Drive around and deliver people places.

This led to a surplus of taxi drivers such that forced taxi drivers to work longer hours to make a living, as the demand and supply curves did not meet at equilibrium. The general public began to grow concerned with the vehicular status of the people operating unofficial taxis, the unregulated environment meaning that dangerous vehicles or drivers could operate freely.

As such, in 1937, the Hass Act was introduced into what became the modern day taxi medallion system.

As for the reasons for creating a medallion system: We may never know 100% exactly what the underlying reasons were behind this addition. Obviously safety will factor into it, but there has been some history of public officials being bribed by the large scale taxi companies, or at least allegations made of these things. Therefore, whether or not the reasons were purely safety oriented is open to debate. Controlling supply and demand, establishing a type of monopoly, these are possible as well.

My personal opinion would be a mix of safety and a wish to have some control over the taxi industry, for both fiscal and economic reasons.

Helpful sources:

http://www.nytimes.com/1996/05/11/nyregion/medallion-limits-stem-from-the-30-s.html

http://nymag.com/guides/everything/taxis/taxi-history-2012-7/

https://books.google.com/books?id=ODnnmcciMLgC&pg=PA461&hl=en