Not to the general population of early 21st century Earth. He can walk down the street without being recognized, unless he runs into someone from UNIT or something.
Within the ranks of UNIT, the upper levels of the government, etc., things are different. We've seen multiple stories where, even if he wasn't directly recognized, once people found out he was the Doctor they were under standing orders (sometimes decades old, or even centuries) to give him special treatment.
The farther in the future you go, the more well known the Doctor seems to be. And by the 52nd century, it seems like everyone knew who he was—but he Doctor fixed that by changing the past and making everyone think he died at Lake Silencio, as he explained in "The Wedding of River Song".
Among the Time Lords, the Doctor was famous, or perhaps infamous, but since they're all gone now, that doesn't matter.
Finally, a few stories imply that he's well known among transcendental super-powered beings. After all, the White Guardian and the Black Guardian, possibly the closest thing to gods in the Whoniverse, fought over him.
Things are pretty much the same in the novels, audios, comics, etc. Normal people don't recognize him on the streets (except briefly in the 1980s, when he's a semi-famous financial genius, but nobody knows that's the Doctor), future human colonies find the Doctor in ancient UN files and decide to listen to him, every Time Lord knows who he is, the Eternals Time and Death both want him as a champion, etc.