After the tax-payers have already paid for a program, it doesn't hurt them if that program also benefits the rest of the world. In fact, if I paid for some television program I would want it to reach as large an audience as possible.
The problem is with all the stuff the BBC doesn't produce itself, but instead licenses from others. Those license agreements are usually much more restrictive than they were when television was simply broadcast to whoever could pick it up. Those radio waves didn't care about national borders, but current licence contracts do.
Hopefully multiple broadcasters in Europe will be able to share the costs of a licence for broadcasting across Europe (or ideally the whole world), so that the total costs for each broadcaster doesn't actually get any higher. Or of course one could just pass legislation specifying what the cost should be, though that probably wouldn't bee free market enough for the EU.