Satellites in ANY orbit are only visible when sunlit. Starlink satellites are in low earth orbit (LEO).
As an astronomer and astrophotographer I can see some satellites -- even quite late -- if they are in a high enough orbit.
LEO satellites are typically visible shortly after sunset or shortly before sunrise because they are not on sunlight very long before passing into Earth's shadow. BUT... this depends on your latitude.
The Earth's axis is tilted roughly 23 1/2° relative to our ecliptic plane. This defines the Arctic and Antarctic Circles (depending on the time of year the Sun never sets). But "astronomical twilight" is defined as the point where the Sun is less than 18° below the horizon (due to the refraction effects of the atmosphere). At your summer solstice, you can add the 18° to 23 1/2° to get 41 1/2° from your nearest pole. This means that for some observers the StarLink satellites may be a problem at any hour of the night ... depending on time of year combined with that observer's latitude.
Having said that ... astrophotographers have been plagued with satellites passing through their images for years. StarLink is just the latest (and by no means, the last) of their issues.
Some stacking (integration) techniques are good at dealing with these problems.
In "stacking" the astrophotographer takes multiple images ... which are ultimately integrated to create a single master image. Images are aligned and then "integrated". In a simplistic integration algorithm, the pixel values for a single given pixel are simply averaged across each sample image. If a satellite or aircraft flies through a just one of many images (suppose it is 10 images) then the averaging method means that the trail of the satellite or aircraft is reduced to just 1/10th of what it would be (since 9 of the 10 sample images did not have the aircraft).
These techniques work well for astrophotographers capturing deep-sky objects. Astrophotographers capturing "nightscape" images (single images of the sky and landscape) do not have multiple sample images that they can use for statistical integration methods ... elimination of light trails from satellites is much more challenging in that scenario.
More sophisticated algorithms ... such as sigma-clipping ... use statistical algorithms to detect outliers. E.g. if 9 of 10 images say an pixel should be "dark" and just one sample images says that pixel should be "bright" then if you consider statistical mean & standard deviation ... 9 samples say the pixel should be "dark" and 1 sample says the pixel should be "bright". Based on statistical methods, the single sample that says the pixel should be "bright" is considered a statistical outlier and the outlier pixels in that sample are rejected ... entirely. This causes satellite trails to COMPLETELY disappear.