Any size tire larger than stock will affect the final drive ratio, and changing gears is the right way to return to "stock" to stay within the engine's power band. Most folks running less than a 33" tire don't bother with new gears, since the effect is slight unless you have something like 3.00:1. There's several charts and formulas to figure what ratio is needed for what size tire, and whether you drive on the street, tow a trailer or want a little better gas mileage. The range for 38's is a 4.88:1 up to a 5.12:1. Different brands of a 38" tire will vary slightly in the actual height, too.
My advice before buying a new set of gears is to research what effects that big a tire will have on your stock axles, transmission, ball joints, etc. A 35" tire is the tallest I would run on the stock Bronco axles, and even that causes more wear and tear on those parts.