Stellar sized black hole?

Depends on what you mean by "influence".... If you mean to alter it's orbital path around the Milkyway, then that could be from *very* far away. If you mean to disrupt the planetary orbits, then I would guess about 1 to 2 light years; If you mean to be considered a binary partner with the sun, then two to 3 light years...

It would have to be MUCH closer than that. Black holes aren't like in the movies. They're not super space sucking vacuum cleaners you know. They're just very dense and very massive. We already have very nearly the equivalent of a 3 stellar mass black hole just barely over 1 parsec from Earth. Alpha Centauri. Alpha Centauri is 2 stars, each of which is almost as massive as our Sun, and Proxima Centauri, which is a red dwarf. As long as you're not close (as in at or near the event horizon) then mass is mass. If our Sun was suddenly transformed into a black hole of its same mass (not possible for such a low mass, still) other than it suddenly getting pretty dark here nothing would change. We would continue on orbiting that black hole like nothing had happened.

I guess that depends on what you mean by "influence" though. Gravity's force diminishes greatly with distance. But, the effects also travel over, as far as we can determine, infinite distance. Pluto's gravity "influences" the Sun. Not enough to notice on a daily basis, but it does. The gravity from your body also "influences" the Earth. Again, not enough to notice.

There is no such thing as a 'stellar-sized' black hole. All black holes, whatever their mass, are of zero size. They are singularities, mathematical points. In fact, that is rather the point about a black hole.

As to how close a black hole would have to be to 'influence' the sun, there is no limit. Gravity works, according to Newton and Einstein, to infinite distances, becoming weaker with distance, but never disappearing completely.