How to take user input as string and check against an array to produce some message ?

A more general solution would be to treat your list of destinations rather like a database and search for the entered text. If you do this then you can extend the list of destinations without having to hard code your test. As a general rule hard coding is a bad thingTM and it can lead to all sorts of unpleasantness including but not limited to code that is fragile and difficult to maintain.

So assuming that the list of destinations is in scope you could do something like the following :

string foundDestination = "";
for (int i=0; i < arrival_location.length; i++) {
  // We'll use indexof but other comparison methods are available.
  // Have a look at String.Equals(...) to start with.
  // There are ways of exiting the loop early when a match is made. 
  // Further reading for you.
  if (arrival_location[i].IndexOf(input, Ordinal) >= 0 ) {
    foundDestination = input;
  }
}

if (foundDestination.length == 0) {
 // Tell the user no match.
} else {
 // Display all matches made.
 // 
} 

If you want to get fancy then there are other ways of searching through data for example...

// Xamarin docs say linq is supported, but I don't know how well.
Using System.Linq;

// We'll use indexof but other comparison methods are available.
// Have a look at String.Equals(...) to start with.
IEnumerable<string> findLocation = From string dest in arrival_location
                                   where dest.indexof(input, Ordinal) >=0
                                   select dest;

// We allow for the possibility of multiple matches
string[] matches = findLocation.ToArray();

if (matches.length == 0) {
 // Tell the user no match.
} else {
 // Display all matches made.
 // 
}

I have also tried something for you, please have a look

class Travel
{


    string[] travellers_name = new string[8];
    public string[] departure_location = new string[4];
    public string[] arrival_location = new string[4];
    int[] mobileno = new int[8];

    public Travel()
    {

        departure_location[0] = "Phuket";
        departure_location[1] = "Amsterdam";
        departure_location[2] = "Mumbai";
        departure_location[3] = "Thailand";

        arrival_location[0] = "1.Bangkok";
        arrival_location[1] = "2.Damascas";
        arrival_location[2] = "3.Sri Lanka";
        arrival_location[3] = "4.Peshawar";
    }
}
static void Main()
{
    string welcome_msg = "Welcome to Sharp Travel Planner .\nPlease select your dream destination\n";
    Travel display_location = new Travel();
    Console.WriteLine(welcome_msg);
    for (int i = 0; i < display_location.departure_location.Length; i++)
        Console.WriteLine(display_location.arrival_location[i]);

    string input = Console.ReadLine();
    int selectedIndex = 0;
    if (!int.TryParse(input, out selectedIndex))
    {
        Console.WriteLine("InValid input!");
        Console.ReadKey();
        return;
    }
    if (selectedIndex > display_location.arrival_location.Count())
    {
        Console.WriteLine("InValid input!");
        Console.ReadKey();

        return;
    }
    var selection = display_location.arrival_location[selectedIndex - 1];

    bool res = display_location.arrival_location.Contains(selection);


    if (res)
        Console.WriteLine("Your choice is {0}", selection);
    else
        Console.WriteLine("Destination not available");

    Console.ReadKey();
}