Marco Alka solution looks good. Also, check out momentjs if you need to deal a lot with dates.

// Save a date. You can pass any date as first parameter to the constructor,
// or just use today's date if you don't pass anything, like below
const savedDate = new Date();

// Create function which checks if date parameter is bigger than saved date
const isFuture = date => new Date(date.toDateString()) > savedDate;

// Get the date from an input field
const dateInput = new Date(document.querySelector('#dateField').value);

if (!isFuture(dateInput)) {
    // Date input is before or on the saved date
    alert('Date must be in the future!');
}
else {
    // work with future date...
}