I see several problems here. First, you're not binding anything to your SQL statement so it's searching for:
SELECT id FROM members where username = ?
It's looking for a user who's name is ?
Next problem, you have your AJAX query malformed. It's sending exactly this:
Which means, your $_GET will = &Username&
This is how the URL should look:
Notice the quote locations? Here, you're telling Construct to add in the contents of the variable Username. You don't need the & at the end unless you're adding something else to the query.
Next problem. You're trying to use the $_GET like an array but you're saying you just want the ID of a specific user. Plus, you're passing the $_GET as a string and you're not 'imploding' it into an array.
And finally, you're not telling MySQL to look for an array of names by using the "IN" statement, you're telling it to use the "WHERE" statement which probably would confuse it and therefore, your $stmt->execute would fail and return nothing.
If you just want the ID of ONE user try this:
$sql = "SELECT id FROM members where username = ?";
$stmt = $base->prepare($sql);
$result = $stmt->get_result();
$row = $result->fetch_object();
Assuming that your usernames are unique, it will return just the first occurence of the $_GET that it finds.