The same way they do now..... They don't!

Pretty interesting to consider. Astrology involves a lot more than most people know. Birth charts are a huge deal for astrologers, and where they make a lot of their money. It takes some work to compile a proper chart and people pay for them.

A birth chart starts using the precise moment and place of your birth. Then the astrologer does a lot of calculations (well, now they just use computers like everyone else, but there are books to lookup and hand calculate this stuff) to determine what celestial bodies were where at your birth.

This is where your sun sign, moon sign, and all the other detailed stuff comes from. And all of that should still be valid (well as valid as it ever was) in a space faring society. Obviously you'd need to use the local view of the sky for determining which planets and constellations to use, and the local superstitions to determine what those things are supposed to mean, but I don't think its so ridiculous to believe that a Star Trek style humanoid alien race which has religion, language, and culture wouldn't also have superstitions and probably, specifically, astrology.

Like how alchemy leads to chemistry, astrology is the prelude to astronomy, and both start with detailed observations of the sky. Sky watching is the basis for time keeping, and most of our advanced math started as a means of tracking time (in order to keep track of holy days, and more importantly, planting days). So i think some form of astrology could well be inevitable on the path to astronomy, physics, sociology, and other real sciences.

As to the problem of daily horoscopes. Those are based on your birth chart but modified by the current sky. That's how you get "daily outlook for all Virgos." In theory, you could just calculate that for the home planet and call it good.

But, we're in the future! On a spaceship! With computers! Why not do complex, subspatial calculus to compute the exact amount of starlight from Proceon 9 bouncing off Mars and reaching the starship you're on? If we assume Trek has solved the three body problem, so that they could calculate how much Venus' gravity affects you anywhere in the universe, why not apply the same "logic" to whatever elusive energies are believed to power astrology?

Your daily space horroscope could include superstitions from numerous cultures in a powerful, exceedingly complex, syncretic universal astronomy.

"So you're a Pisces? Not good, from our current coordinates, Uranus is occluded by the second sun of Chronos, you'll find it hard to focus on relationships while the Klingon warrior spirit is filtering all your energy. I recommend embracing this, drink a raktajino, and channel the fighter's energy into your work and overcome these obstacles. Q'pla!"

Not too complicated. There are already LLMs spewing that nonsense.

As long as we're talking an actual horoscope, not the lazy bullshit that you find in newspapers, you have a large number of variables with intricate relationships and various degrees of importance. These can be interpreted in various ways - often completely opposite, meaning any given configuration can fit most if not all situations of one's life. This is the same for other kinds of divination - tarot, palm reading etc.

The trick to reading it is to focus your conscious mind on the rules and leave your subconscious to pick among the ambiguities according to a deeper understanding of your situation free from conscious biases (or, when reading for others, picking up on subtle cues that you wouldn't notice or interpret correctly consciously). You basically lean into the illusion that you're reading the horoscope in the one possible way according to the rigid rules while you're actually tapping into your subconsciousness to pick on things that would otherwise either not occur to you or which you'd rationalise away.

It works surprisingly well even if both parties are aware that this is the approach the reader is taking, much like the placebo effect works even on people who know what placebo is. It will, of course, not tell you about future events, but it's a great tool to look into a particular situation, work out your feelings about it and perhaps even suggest a good way of dealing with stuff.

Anyway, in space it would work exactly the same as on Earth since the input data is largely irrelevant to the divination, though being aware of the extra impossibility could affect the reading, so if you find yourself in deep space, I'd definitely recommned using tarot instead.