How can I check if an object is an array?

You can use Array.isArray(). Here is a polyfill:

if (Array.isArray == null) {
  Array.isArray = (arr) => Object.prototype.toString.call(arr) === "[object Array]"
}

I would make a function to test the type of object you are dealing with...

function whatAmI(me){ return Object.prototype.toString.call(me).split(/\W/)[2]; }

// tests
console.log(
  whatAmI(["aiming","@"]),
  whatAmI({living:4,breathing:4}),
  whatAmI(function(ing){ return ing+" to the global window" }),
  whatAmI("going to do with you?")
);

// output: Array Object Function String

then you can write a simple if statement...

if(whatAmI(myVar) === "Array"){
    // do array stuff
} else { // could also check `if(whatAmI(myVar) === "String")` here to be sure
    // do string stuff
}

If the only two kinds of values that could be passed to this function are a string or an array of strings, keep it simple and use a typeof check for the string possibility:

function someFunc(arg) {
    var arr = (typeof arg == "string") ? [arg] : arg;
}