Well, I imagine there's two issues. One, the amount of people who know both D&D and have more than a cursory knowledge of astrology is pretty small. Two, the fact that there are 12 classes and 12 signs means you don't want to do just a random response, you'd want to do a more holistic selection where you assign each class to exactly one sign.
That being said, let's try it!
Each sign is associated with one of the four classical elements, so let's start there.
Air signs (Gemini, Libra, Aquarius) are typically intellectual and communicative, so classes focused on Intelligence or Charisma would make sense here. Wizard and Bard are definites.
Water signs (Cancer, Scorpio, Pisces) are empathic and sensitive, so Wisdom classes seem a fit here. Cleric and Monk, and maybe Druid or Ranger.
Fire signs (Aries, Leo, Sagittarius) are aggressive and zealous. I'd lean towards hard hitters and Charisma focused classes here. Sorcerer almost definitely, maybe paladin and barbarian.
Earth signs (Taurus, Virgo, Capricorn) are pragmatic and stubborn. Less magical and more survivable classes would fit here, like Fighter.
I'll look to the ruling planets for tiebreakers. Ruling planets are as follows, and tend to have the thematic associations you would imagine for each planet.
Aries - Mars
Taurus - Venus
Gemini - Mercury
Cancer - Moon
Leo - Sun
Virgo - Mercury
Libra - Venus
Scorpio - Mars (or Pluto in some systems)
Sagittarius - Jupiter
Capricorn - Saturn
Aquarius - Saturn (or Uranus in some systems)
Pisces - Jupiter (or Neptune in some systems)
So let's go from options I feel are pretty obvious, then wrestle with the less obvious options.
Wizard - Gemini. Air sign, Mercury ruling. Obvious focus on intellect, and Gemini is known for being changeable and flexible, much like wizard spell prep.
Bard - Libra. Air sign, Venus ruling. Extroverted, communicative sign fits perfectly with bard.
Fighter - Taurus. Stubborn and implacable sign, fits well with the durable fighter.
Sorcerer - Leo. Fire sign, ruled by the Sun. Leo is a sign of potent resolve and tons of drama, which fits the energy and presence of a sorcerer.
Paladin - Aries. Another fire sign, but more aggressive and idealistic than Leo. Fits with the drive and zealousness of the paladin.
Barbarian - Sagittarius. Bold, adventurous, and a risk taker. Jupiter as the ruling planet implies an outsize presence and a lust for life.
Rogue - Capricorn. Pragmatic and strategic, always trying to be in control and five steps ahead.
Cleric - Cancer / Druid - Pisces. Tough call between Cancer and Pisces for these two, but Cancer is community oriented, while Pisces is more big picture and dreamy. Pisces being a mutable sign also fits the form shifting druid.
Ranger - Virgo. Practical and service oriented fits the class conception of the ranger. Mercury ruling implies a more intellectual bent, fitting the ranger as a master of wilderness lore.
Monk - Scorpio. Tough call between Monk and Warlock for Scorpio. Scorpio is traditionally a darker, emotional sign (ruled by aggressive Mars and fatalistic Pluto), but its inner focus and more spiritual bent make me lean towards Monk.
Warlock - Aquarius. A sign focused on innovation and breaking convention, it seems an appropriate sign for a class that will break any taboo in order to gain more power and knowledge.
