Since your more thorough validation should be on the server-side anyway, you could just use parseInt or parseFloat depending on what sort of value you are expecting. Then check if the result is actually a number and that it also meets your constraints:
var number = parseFloat($('.billboard-height').val()); // or parseInt depending on expected input
if (isNaN(number) || number < 150) {
$('.sb-billboardalert').fadeIn(600);
}
EDIT:
Based on your comments, you are entering regex land. I gather you only ever want a natural number (and the way parseInt/parseFloat ignores trailing non-numeric characters like px, em, etc. is not ok). How about:
var val = $('.billboard-height').val();
var number = parseInt(val, 10);
if ( ! val.match(/^[0-9]{3,4}$/) || number < 150) {
$('.sb-billboardalert').fadeIn(600);
}
This should only allow natural numbers 150-9999.