Your question is difficult to address for a few reasons. By asking "what time did this get dangerous...", we’re assuming that parents started keeping a closer eye on their kids because things got scarier than they were in the 80’s. However, over the last few decades, violent crime rates have actually dropped significantly. Violent crimes peaked in the early 90's and had been growing steadily through the 1980's. By comparison, both violent and property crime rates in the 1980's were significantly higher than they are currently.^1

So, its important to note that the "public perceptions about crime in the U.S. often don’t align with the data.".^2 So for me, the question becomes, why were parents more comfortable in the 1980's giving their children a larger amount of freedom? There are certainly a variety of variables you could investigate. The common want to blame media coverage would be one place to start, but probably doesn't offer the full story.

Potentially, with crime rates peaking in 1992 and Bill Clinton passing his Crime Bill in 1994, the topic would be on people’s minds.^3 This in combination with media coverage of the crack cocaine epidemic could be enough for parents of the 90’s to alter their parenting methods. As for why the average American parent is not comfortable giving their children similar freedoms now that crime is significantly lower than it was in the 1980’s, I do not know and hopefully someone can chime in. However, I don’t even know if that is entirely true because the premise itself seems anecdotal and I couldn’t find any concrete sources to verify whether kids are more or less independent in today’s America.