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The graph on page 17 (PDF page 18) of this Boeing air safety publication shows the fatal accident rate for commercial jets in the US and Canada dropped from about 40 per million departures in 1959 to 2 per million departures in 1962. Since then, the rate has declined even further, but nothing as dramatic as the original drop.
What happened during the the 1959-1962 period to so greatly reduce the accident rate?
I believe the answer has something to do with the introduction of the passenger jet airliner, which first came into service in the late 1950s. Note please that the original poster is asking for information on "commercial jets" whereas the statistics that others have offered have information on commercial planes in general, including prop planes. I glanced at the statistics for 1959 presented above and there were only two passenger jets (including one KC-135, which you may discount if you like) that crashed.
Compare this to the statistics presented for 1962. There were six or seven crashes of Boeing and Douglas jet airliners.
To me this means the differences were in all probablility not statistically significant. We don't have enough "samples" to reach a valid conclusion.
What you can see is a general explosion in the number of people flying. Jets really introduced the public to flying.
No doubt, prima facie what's most pertinent needs to be stated at the outset. There is an element of randomness to all air crashes.
That said, sustained efforts by US and some foreign airlines to identify and eliminate minor problems that generally are common precursors to accidents brought down fatality rate drastically from the '60s onwards.
One of the most common scenarios for a plane crash in the '50s was Controlled Flight Into Terrain(CFIT) referring to aircraft that were piloted into the ground, water, mountains or other terrain. However, improvements in cockpit instruments and jet engines dramatically improved aircraft performance and brought down the number of CFITs in the early '60s.
Look at the number of air crashes in 1959, 1960, 1961, 1962. The number of air crashes in the US fell from around sixteen in 1959 to around four in 1962. (Check above link for exact figures.)
Moreover, every single air crash that was successfully investigated provided data to the OEMs that was used to improve the aircraft. For example, the public inquiry into the Comet airliner disaster in 1954 found out that "Metal Fatigue" was the cause of the accident. The investigators found that a small weakness such as this would quickly deteriorate under pressure, and would rapidly lead to a sudden and general break-up of the fuselage.
http://news.bbc.co.uk/onthisday/hi/dates/stories/october/19/newsid_3112000/3112466.stm
Look at this image below. It reflects the causes of air accidents:
You will notice that some of the major causes of air accidents like Pilot Error and Mechanical Failure have gone down between the '50s and '60s.
http://www.planecrashinfo.com/cause.htm http://www.thepoliticalindian.com/plane-crash-reasons-since-1950/
Lessons were learnt from past accidents. Consequently, companies like Boeing brought about necessary improvements to the avionics and engines. Airliners also helped pilots hone their skills. A combination of these two factors brought down the number of air crashes.
Even to this day, OEMs and airlines are making effort to further improve flight safety.
One oft-cited example is a discovery in the last decade by US Airways (then US Air) that many of its planes approaching Charlotte/Douglas International Airport in North Carolina were coming in “high and hot” - too fast and at a steep angle. After the investigation into this issue was completed, the FAA changed the approach procedure there and the airport installed a system to guide planes at a proper angle.
http://www.ntsb.gov/safety/safety-recs/recletters/A-13-024.pdf
When you are looking at "per capita" style numbers you also have to look at the overall volume in your population. Given that the time period is the general start of the growth of aviation, I'm going to guess that you would find that overall flights (and passengers, miles, etc, etc) increased during the same period. That would make the "per X" number of virtually anything you look at in aviation improve.
