Passing a method parameter using Task.Factory.StartNew

The best option is probably to use a lambda expression that closes over the variables you want to display.

However, be careful in this case, especially if you're calling this in a loop. (I mention this since your variable is an "ID", and this is common in this situation.) If you close over the variable in the wrong scope, you can get a bug. For details, see Eric Lippert's post on the subject. This typically requires making a temporary:

foreach(int id in myIdsToCheck)
{
    int tempId = id; // Make a temporary here!
    Task.Factory.StartNew( () => CheckFiles(tempId, theBlockingCollection),
         cancelCheckFile.Token, 
         TaskCreationOptions.LongRunning, 
         TaskScheduler.Default);
}

Also, if your code is like the above, you should be careful with using the LongRunning hint - with the default scheduler, this causes each task to get its own dedicated thread instead of using the ThreadPool. If you're creating many tasks, this is likely to have a negative impact as you won't get the advantages of the ThreadPool. It's typically geared for a single, long running task (hence its name), not something that would be implemented to work on an item of a collection, etc.

class Program
{
    static void Main(string[] args)
    {
        Task.Factory.StartNew(() => MyMethod("param value"));
    }

    private static void MyMethod(string p)
    {
        Console.WriteLine(p);
    }
}