To be fair, modern day stories have decreased somewhat since Moffat took over. RTD really enjoyed modern-day episodes. Probably just as much because he's a fan of Pertwee as it was it being cheaper.
I think it's also, to a large extent, because RTD was basically introducing
Doctor Who to a whole new generation of viewers, and so he chose to ground it in the familiar present-day world so as not to scare off the newbies. Just as it was also more grounded on a character level, touching more on ordinary, everyday concerns of working-class council-flat residents and the like.
There's also the fact that he was heavily influenced by
Buffy the Vampire Slayer, so he tended toward "weird stuff happening in the regular world" over traveling to exotic parts of the universe.
For that matter, even when we do see the future in modern Who, regardless it it's RTD or Moffat era, it's often very similar to modern-day. EG, everyone still wears contemporary outfits. Human military personnel wear uniforms which wouldn't look out of place today, their weapons and gear often are modern ones, and so on.
Just as on the reboot
Battlestar Galactica, the natives of an alien civilization wore ordinary clothes and such. I find that audiences today are depressingly intolerant of exotic costumes, designs, and the like in their sci-fi. Shows in the '60s and '70s were able to be wilder with the designs, although they're generally considered cheesy and dated-looking today.
But to answer the question I'm not sure how production should handle it. One of the Doctor's characteristics is that he generally becomes the figure in charge when he inserts himself into a situation. They could continue with that or maybe sometimes he'd have trouble establishing his authority due to his race.
But wouldn't a white Doctor face that same issue in a story set in, say, Tokugawa Japan or Maurya-Dynasty India or the Empire of Mali? This was something the original series glossed over quite a bit in serials like "Marco Polo" and "The Aztecs," but the white travelers would've been the racial minorities in those cultures; indeed, the Aztecs would never have seen anything like them before. And realistically, if the Doctor's visits to Earth were anywhere near random in their geographical distribution, he'd land in Asia and Africa much more often than he landed in Europe. So if anything, the fact that the Doctor looks white
should be an impediment to him much more often than it's shown to be.