How can I make sure the User can input again after entering something invalid?

Use this method to validate the input.

public int validInput() {
    Scanner scanner = new Scanner(System.in);
    while(1) {
        // If you wish to end the loop after a certain number of attempts
        // use a counter to exit the loop
        System.out.println("Enter the number:");
        int num = scanner.nextInt();
        if(num > 0) {
            return num;
        }
        System.out.println("Input must be greater than zero. Please try again.");
    } 
}

Then call this method from your code.

while (cont == 'y') {
  int num = validInput();

  int factorial = fact(num);
  System.out.println("Factorial of entered number is: " + factorial);

  System.out.println("Do you want to loop again?");
  cont = input1.next().charAt(0);
 }

You can also use this method anywhere in your project.

Do it as follows:

import java.util.Scanner;

public class Main {
    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);
        char cont = 'y';
        int num;
        boolean valid;
        String strNum;
        do {
            valid = false;
            System.out.print("Enter the number: ");
            strNum = scanner.nextLine();
            if (strNum.matches("\\d+")) {
                int factorial = fact(Integer.parseInt(strNum));
                System.out.println("Factorial of entered number is: " + factorial);
                System.out.print("Do you want to loop again?");
                cont = scanner.nextLine().toLowerCase().charAt(0);
            } else {
                System.out.println("Invalid entry. Please try a positive integer.");
            }
        } while (cont == 'y');
        System.out.println("Goodbye!");
    }

    static int fact(int n) {
        int output;
        if (n == 1) {
            return 1;
        }
        output = fact(n - 1) * n;
        return output;
    }
}

A sample run:

Enter the number: a
Invalid entry. Please try a positive integer.
Enter the number: 10.5
Invalid entry. Please try a positive integer.
Enter the number: -5
Invalid entry. Please try a positive integer.
Enter the number: 5
Factorial of entered number is: 120
Do you want to loop again?y
Enter the number: 3
Factorial of entered number is: 6
Do you want to loop again?n
Goodbye!

Notes:

1. Use of do...while is more appropriate in this case. It doesn't mean that it can not be done using while loop but do...while makes it more comprehensible.
2. Compare scanner.nextLine().toLowerCase().charAt(0) with y so that the program can continue for both Y and y.
3. \\d+ is used to restrict a string matching with only digits and nothing else i.e. it allows only intgers.
4. Integer::parseInt is used to convert an integer string into int.
5. You do not need two Scanner instances.
6. Follow Java naming conventions e.g. class recursion2 should be named as class Recursion2 as per the naming convention.

To do this, you need a do-while loop where you check each time if the input is valid, and also a try-catch block inside the loop so if the Scanner throws an exception, you can catch it and ask the user to enter a positive number.

Also, you need to close the scanner at the end.

import java.util.Scanner;
 class recursion2{
 public static void main(String args[]){

   Scanner input1 = new Scanner(System.in);
   Scanner scanner = new Scanner(System.in);
  char cont = 'y';
 while (cont == 'y') {

  System.out.println("Enter the number:");

  int num = -1;  //Negative by default so it's invalid
  do {  //Do-while so it gets run at least once
    try {
      num = scanner.nextInt();
    } catch (InputMismatchException e) { //When it's not an int, loop again
      System.out.println("Please enter a valid input");
    }
  while (num <= 0); //Check if positive here


  int factorial = fact(num);
  System.out.println("Factorial of entered number is: " + factorial);

  System.out.println("Do you want to loop again?");
 cont = input1.next().charAt(0);
 }
 scanner.close();
 }
 static int fact(int n)
 {
   int output;
   if(n == 1){
     return 1;
   }

   output = fact(n - 1) * n;
   return output;

 }
 }