This is kind of a broad question so I'll impose a few limitations of my own here. Partly to make it a little narrower in scope but also because my knowledge has limits.

First of all "people" is a pretty broad term. Some people have been running for a very long time. The leisure classes of many societies have competed in foot races for quite literally thousands of years and since they were, in general, not forced to do so, one would imagine that quite a few of them did it for what might be called leisure. However, just because some ancient greeks, romans and people of other cultures ran regularly it does not follow that it was a popular activity with the broad masses.

Second, I will speak about western Europe and the USA here. I know absolutely jack about the history of running in any other part of the world.

Third, while I will not go into depth about the history of aerobic fitness as a part of overall military fitness there is no deny that there is a connection there. The advantages of aerobic fitness have been know for a very long time in what might loosely be called the military establishment. However, I don't believe anyone who has done regular running with a drill sergeant would refer to it as a leisurely activity. It certainly has health benefits though.

So, I will try to answer a question close to what you asked. Something like "When did running gain mainstream popularity in western Europe/the US?".

The TL;DR is during what is called the Running Boom of the 1970's.

That said, it's not as if millions of people suddenly got out of bed on new years day in 1970 and went "Right, let's go jogging". As with all such cultural shifts it is nearly impossible to pin it to a single defining moment. We can however look at a few things that were certainly influential in making running a popular activity for the masses.

One thing that certainly played a part was the dramatic events in the 1972 Olympic Marathon in Munich. American runner Frank Shorter took home the gold, the first time this happened in a televised event. However, a German student in West German track uniform actually crossed the finish line first. He was however an imposter who entered the track just outside the stadium. Something officials quickly realized. Somewhat sadly for Shorter he entered the stadium to boos and jeers he initially thought were directed at him.A somewhat interesting side note is that while this was the third time a US runner had won the men's marathon, someone else had crossed the finish line first in all three cases. In 1904 Fred Lortz (US) crossed the finish line before Thomas Hicks after having traveled much of the distance by car and in 1908 Johnny Hayes finished after Italian runner Donardo Pietri. However, Pietri was so exhausted and disoriented that when he entered the stadium that he took a wrong turn and fell several times. He required help to reorient himself and get up and was disqualified after a protest from the US team.

So, Frank Shorter got people excited about running but he did not start the trend. It's very hard to pinpoint who did. Jack Lalanne had already been an influential figure in popularizing fitness in general for many years at this point. It is perhaps impossible to separate the two. Without the rising popularity of fitness in general I doubt the running boom would have been as impactful as it was. When Shorter won the gold in Munich Lalanne had already been promoting fitness for 19 years on national television (for more on Jack Lalanne and the popularization of fitness, see this thread by me, /u/superplaner, I see you automoderator).

Another thing we have to consider is the massive mobilizations and preparations that were undertaken by many nations in the 1930's and 40's. A generation of, primarily, men were forcibly introduced to aerobic exercise. It was a field that was studied quite intensely by scientists in the years leading up to that. Archibald Hill introduced vO2-max in the early 20's and recieved the Nobel Prize for his work on muscle metabolism in 1922 along with Otto Meyerhof. Several scientist continued to build upon the work of Hill and Meyerhof in the following decades leading to a better understanding of the science behind it. Many of the exercise programs originally developed for the armed forces became very popular in the post war years. In America most notably the Canadian Airforce Exercise plans (5BX and XBX) which both have an aerobic component.

Inventions such as the cushioned running shoes probably also deserve a mention but I know very little about it. I find it hard to imagine that running would have become as popular is it did with uncushioned shoes.

So where does that leave us? Well. It leaves us with a gradually increasing interest in fitness that started some time in the early 20th century. That, along with increased scientific understanding of aerobic exercise, better equipment and a few popular figureheads combined to make running a popular activity with the broad masses in the 1970's.

I realize I left out a lot here. I want to talk about Browning Ross, the founder of the Road Runners Club of America. About Bill Bowerman and his influence of training and equipment. I want to mention the Flying Finns. I left out Kathrine Switzer and Miki Gorman and at least half a dozen female pioneers who helped make the sport popular among women (Gorman in particular given how late she started, that had a great influence in that it showed a lot of people that you didn't have to start in high school to get good). I left out so much but the post is long enough as it is. Feel free to hit me up if you have follow-up questions.