Get a specific value from a big JSON response
Since Swift 1.2, multiple optional bindings and boolean conditions
can be combined to a single if-statement. This helps to avoid the
"Swift if-let pyramid of doom". In your case that would be

if jsonResult.value(forKey: "status") as! String == "OK",
    let routes = jsonResult.value(forKey: "routes") as? NSArray,
    let legs = routes.value(forKey: "legs") as? NSArray,
    let duration = legs.value(forKey: "duration_in_traffic") as? NSArray {

    for i in 0..&lt;duration.count {
        let timeDuration = duration[i] as! NSArray
        if let time = timeDuration.value(forKey: "value") as? NSArray {
            print("Current Time: \(Int(time[0] as! NSNumber)/60) Min")

        }
    }
}


and reduces the maximal nesting level by three.

There are two "forced unwraps" in your code:

let jsonResult = try JSONSerialization.jsonObject(with: data, options:JSONSerialization.ReadingOptions.mutableContainers) as! NSDictionary


and 

if jsonResult.value(forKey: "status") as! String == "OK",


which can cause a program termination at runtime. Better use
optional casts. In the second case that would simply be

if jsonResult.value(forKey: "status") as? String == "OK",


because optionals and non-optionals can be compared. 

The JSONSerialization.ReadingOptions.mutableContainers option
is not needed because you are not modifying the returned dictionaries
or arrays.

You are using value(forKey:) a lot. This is a method from the
Key-Value Coding protocol, and is often not what you need. In

let routes = jsonResult.value(forKey: "routes")


you are simply retrieving a dictionary value, that is:

let routes = jsonResult.object(forKey: "routes")


In 

let legs = routes.value(forKey: "legs")


you are mapping the routes array to a new array legs. This is
Key-Value coding, however it can be confusing, in particular in this
case where legs would be an array of arrays. This makes duration
and timeDuration nested arrays as well, and is the reason why
you have to subscript time[0] finally. If you are interested
in the first route only, then it is easier to extract the first
element of routes and continue to work with that.

Generally it is preferred to use native Swift type instead of
Foundation types, e.g. [Any] (or a more specific type) for an array,
or [String, Any] for a dictionary. Compare 
Working with JSON in Swift from the Swift blog for more information.

Instead of

for i in 0..&lt;duration.count {
    let timeDuration = duration[i]
    // ...
}


you can simpler write

for timeDuration in duration {
    // ...
}


Putting it all together, I would write the code like this:

let jsonResult = try JSONSerialization.jsonObject(with: data)

DispatchQueue.main.sync {
    if let jsonDict = jsonResult as? [String: Any],
        jsonDict["status"] as? String == "OK",
        let routes = jsonDict["routes"] as? [[String: Any]],
        let route = routes.first,
        let legs = route["legs"] as? [[String: Any]] {

        for leg in legs {
            if let duration = leg["duration_in_traffic"] as? [String: Any] {
                if let time = duration["value"] as? Int {
                    print("\(time / 60) Min")
                }
            }
        }
    }
}


Note how duration["value"] can be conditionally cast to  an Int
without an intermediate NSNumber.

Finally you can factor the extraction of the duration data into
a separate function:

func durations(from directions: [String: Any]) -&gt; [Int]? {
    guard directions["status"] as? String == "OK",
        let routes = directions["routes"] as? [[String: Any]],
        let route = routes.first,
        let legs = route["legs"] as? [[String: Any]]  else {
            return nil
    }

    return legs
        .flatMap { $0["duration_in_traffic"] as? [String: Any] }
        .flatMap { $0["value"] as? Int }
}


which can then be used as

if let directions = (try JSONSerialization.jsonObject(with: data)) as? [String: Any],
    let durations = durations(from: directions) {
    DispatchQueue.main.sync {
        for time in durations {
            print("\(time / 60) Min")
        }
    }
}


Here flatMap() is used to map each "leg" to its duration,
this is essentially what you did with value(forKey:) in your 
original code.