You can loop over the arrays, see how many are in each one, then filter out the ones that don't have any, then count them. There are lots of ways to do this as others have mentioned. I think all the answers show different methods of how you could do it. You should familiarize yourself with all the methods available to Arrays. https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array
Here is another option with explanation:
let arraysWithMoreThanOne = Object.keys(playerByGender).map(function(key) {
// starting with an array of keys men, women, other
// the map method allows us to convert the key into something else.
// in this case, the length of the array
// we only care about the lengths
return playersByGender[key].length;
}).filter(function(count) {
// filter allows us to discard item in an array we don't care about
// we only care if they have more than 1 in them
return count > 0;
}) // at the end, we have converted the object into a different result
.length; // this is the number of arrays that had more than one item
Depending on your environment, some methods need polyfills to support.