Assuming no artificial light, what is the minimum number of degrees apart would the sun and Venus have to be during the beginning of civilian twilight for the planet to be visible to the naked eye to the casual observer.
Around 15 degrees.
Viewing Venus when it's near the Sun is difficult, for much the same reasons that it's difficult to view the Moon when it's a very thin crescent. You have to deal with the light of the sunset, and when viewing celestial bodies near the horizon you're looking through a lot of air.
According to the United States Naval Observatory (USNO),
Generally, the lunar crescent will become visible to suitably-located, experienced observers with good sky conditions about one day after New Moon. However, the time that the crescent actually becomes visible varies quite a bit from one month to another. Naked-eye sightings as early as 15.5 hours after New Moon have been reliably reported while observers with telescopes have made reliable reports as early as 12.1 hours after New Moon. Because these observations are exceptional, crescent sightings this early in the lunar month should not be expected as the norm.
The apparent area of the crescent also increases inversely with the square of the distance to the Moon. Some of the earliest reliable sightings of the crescent occur near elongations of around 10 degrees. Simply specifying the age or elongation of the Moon cannot tell the whole story. But the elongation is a more reliable parameter to use as a starting point in assessing the lunar crescent visibility at any given date and time.
There are two circumstances where Venus appears close to the Sun: superior conjunction, when it's on the opposite side of the Sun to Earth, and inferior conjunction, when it's between the Sun and Earth. Obviously, it will appear largest when it's near Earth. But it won't be at its brightest at that time because it will be a very thin crescent.
According to Graham Jones from timeanddate at superior conjunction Venus is hidden for ~50 days. At inferior conjunction, it's hidden for ~8 days. But those numbers must be for telescope observation. For naked-eye observers, we need to add a few days.
The most recent inferior conjunction of Venus was on August 13, 2023. Here's a telescope photo of Venus, courtesy of earthsky.org, taken by Roberto Ortu in Cabras, Sardinia, Italy, on the 3rd of August 2023 at 16:08 local time (4h 27m before sunset).
At that time, the elongation of Venus from the Sun was ~17°, and although it was 10 days before conjunction, the crescent is very thin.
Here's a plot produced using JPL Horizons of the Venus-Sun elongation around the time of the conjunction. (The timestep is 1 day, with points plotted at 0:00 UTC).
According to that EarthSky article, Venus was expected to be visible as the Morning Star by the 21st of August, when its elongation was ~14°.
The Moon is obviously brighter than Venus. However, per unit angular area, Venus can be brighter than the Moon. That's because the Moon is made of rock, with the mean albedo of aged asphalt, whereas Venus is covered with dense clouds. One morning I was watching Venus and a very thin crescent old Moon, and the Moon became invisible to me a good hour before I could no longer see Venus.
Under ideal conditions, when Venus is near its maximum brightness, and near the Moon so that it's (relatively) easy to locate, Venus can be seen at any time during the day. About 10 years ago, under such conditions, I saw Venus at every hour of the day that it was above the horizon, even at noon. Admittedly, the weather was excellent, and I was living near the coast, ~20 km from the nearest large town, so there was no air pollution.
FWIW, here are some Horizons plots for a recent synodic cycle of Venus, between two superior conjunctions. (The timestep is 7 days, points plotted at 0:00 UTC).
Note that the magnitude and surface brightness are only approximate. From the Horizons docs:
Magnitudes are, in principle, accurate to about +/- 0.1 magnitude. However, measurement and calibration issues mean values should be considered uncertain at the +/- 1.0 magnitude level.