What makes this somewhat different form the case of Mars is that the orbit of Venus is pretty round. It has even less eccentricity than the orbit of the Earth.
Therefore, all "good" close approaches are going to happen close to the perihelion of the Earth.
As a first order approximation, that means "good" encounters are somewhere in the range between the difference between Earth perihelion and Venus perihelion and aphelion, 39.62 and 38.16 million kilometres.
There haven't been too many such "good" encounters since the beginning of the space age, so from tabular values, the closest one would be the one in January 2014, at 39.811 million kilometres (solex printout).
(The one in January 2022 is going to be 0.05 million kilometres closer, so you better start planning now if you want to grab the record for yourself)
Most of this range can be cut away by considering relative inclination and argument of perihelion.
What remains after that is the same wildcards as for Mars:
- Up to 0.4 million kilometres reduction if any of the Apollo missions happened during a close encounter. (But the same solex printout shows there were not good encounters between 1946 and 1990)
- Up to 0.0004 million kilometres reduction if the ISS has a good beta angle.
- Up to 0.00001 million kilometres reduction if any plane was in a good spot.
- Up to 0.000008 million kilometres reduction for mountain climbers. (Unlike Mars, this would happen at noon).
- Up to 0.000000001 million kilometres by jumping into the air.
Subtracting the radius of the Earth and Venus, that's 31.797 million kilometres.
The greatest uncertainty left is only 400km.