Culex is a superboss in Super Mario RPG, and can be quite difficult with his (and his crystals') wide array of strong attacks, many of which are unblockable. Here is what I did to defeat him!
(Note: I fought Culex in the Nintendo Switch remake of the game - this strategy won't work as well in the SNES version, but I will include some potential alternative strats where necessary! I'll also be using the Switch version's item and spell names.)
Prerequisites
Culex can be found in Monstro Town, two doors across from Monstermama's home. In the Switch remake, this door is an ominous black in contrast to the regular doors' grey. In order to fight Culex, you will need the Shiny Stone in order to break the seal on the door leading to his encounter. You can get this stone from a girl in Moleville running a pretend store, who will exchange the Shiny Stone for some fireworks. The fireworks can be obtained from another NPC in Moleville for 500 coins.
Levels, Items and Equipment
At the time I made my attempts at the Culex fight, my party was at level 20, and I had just gotten the sixth Star Piece. Levels can be slightly higher or lower depending on your preferences, but if you want to bring the Quartz Charm into the final area of the game, I would advise not making any further story progress. Do Not get on that bus!!!
For recovery items, I stocked up on as many Mid Mushrooms, Maple Syrups and Pick Me Ups as I could. You're going to want to be able to recover your FP so you can use powerful moves as often as possible, and Pick Me Ups just in case something goes horribly wrong. I also bought a few Finless Colas from the Juice Bar item shop at Tadpole Pond, but if you have the coins, stock up on a few Croaka Colas instead, or maybe a mix of both. Croaka Colas also restore FP, which could come in handy in a pinch. Finless Colas are only available if you have the Tenor Card, and Croaka Colas are only available if you have the Soprano Card, both of which can be obtained helping Toadofsky at Melody Bay 2/3 times respectively. Better get looking up the solutions!
Also, if you're a Monster List nut like me, I recomend having at least 15 Frog Coins in your pocket after the fight. This is not a fight you want to be Thought Peeking in.
Equipment was trickier for me to figure out, but I did eventually land on something that worked for me. Obviously, everyone should have their strongest weapons and armour equipped, but the accessories were a bit of a struggle.
Mario got the Trueform Pin, which prevents being turned into a Mushroom or a Scarecrow. The Wind Crystal packs Petal Blast, which turns the whole party into Mushrooms if it hits. Mario's a well-rounded character stat-wise, so having him essentially locked out of battle until healed is problematic in a fight that's blasting you with powerful spells every turn.
(Note: These equipment screenshots were taken after my fight with Culex for this strategy writeup. Party levels are slightly off from what I went into the fight with, as I wasn't planning to make this before I did the fight. Mario, Mallow and Peach went into the fight one level lower.)
Mallow wasn't a character I wasn't using actively in this fight, so he kept the Wake-Up Pin I put on him hours in the game earlier. It prevents Mute and Sleep, which is helpful for a spellcaster like himself. The Wind Crystal can inflict Sleep with Light Bubble, so it is something to keep in mind.
Geno got the Ghost Medal, which doubles defense in battle. This stat change doesn't show up in the menu, but Geno wasn't dying as often, so it works! You can get the Ghost Medal by completing a short sidequest activated by sleeping in Monstro Town's inn-in-all-but-name, in which a trio of undead (but friendly) monsters calling themselves the Three Musty Fears will challenge Mario to find the three invisible flags they've hidden somewhere in the world. These three flags are located in Mario's Pad, in Rose Town and on Yo'ster Isle. Once you've found them all, simply sleep in Monstro Town's "inn" again and the Three Musty Fears will equip the Ghost Medal onto Mario.
Bowser can come in handy during the Culex fight, but he wasn't part of the active party for most of it, so I gave him Booster's Charm, which boosts all stats but lowers speed by 5. Bowser's the slowest party member anyway, so I wasn't super worried about this. If you're doing this fight at the same point I did, then the only chance you had to get Booster's Charm was by successfully hiding from the Snifsters at the top of Booster's Tower. If you don't have Booster's Charm, then you could alternatively equip the Zoom Shoes (giving Bowser lower Attack and Magic Attack stats than he would with Booster's Charm, but only slightly less Defense and a whole lot more Speed) or defeat Jinx three times at his dojo in Monstro Town to receive the Jinx Belt (much higher Attack, Defense and Speed, but also much lower Magic Attack/Defense stats).
Peach is a very important party member for this fight, so you'll want to keep her protected from statuses so she can heal others when her turn comes around! I equipped the Safety Badge onto her earlier in the game, and I kept it on for the Culex fight.
The Fight
I opened this fight with Mario, Bowser and Geno. Bowser and Geno can do a lot of non-elemental multi-target damage, which is great for this fight. If you're playing the SNES version, I would substitute Bowser with Peach, as you will need her sooner or later. If you're playing the Nintendo Switch version, it's helpful to have at least some charge in your Action Gauge. I had 44% in mine at the start of the battle. If you're good at Action Commands, you'd be fine having even less in there.
Culex's first turn will be spent on doing a short speech and then summoning the four crystals that assist him in the battle - and this can be outsped. I, at one point, equipped the Feather accessory onto Geno, giving him a whopping 50 speed and making him move before Culex. This can be handy if you want to get a potshot off on Culex himself, but I wouldn't recommend it. The crystals are the more important targets, ultimately. That attempt was also my worst one.
Do as much damage to as many targets as you can on your opening turns. I personally used Geno Flash, Mechakoopa Stomp (if Bowser's active)/Psych Bomb (if Peach is active) and Ultra Jump for this. Don't bother trying to afflict a status condition onto anyone, as they're all immune to all statuses. Geno Flash and Mechakoopa Stomp do non-elemental damage, which as mentioned before, is fantastic to have in this fight. Every crystal has its own elemental weakness and resistance, which can be quite bothersome. Of all my active party members, Mario is afflicted the worst by this, as both of his multi-target attacks have an element associated - Jump for his jumping moves and Fire for his fireball moves. No matter which you use, one of the crystals is going to take no damage from Mario's attacks. The Earth Crystal resists Jump, and the Fire Crystal resists (wouldn't you know it) Fire, but the Earth Crystal has more HP, more unblockable attacks and can inflict Fear. So, retrospectively, I'd suggest using Ultra Fireball with Mario instead, unless you'd like to be slightly more conservative with your FP. If that's the case, you may also want to consider using Geno Blast as well.
When the crystals attack, there is a high change someone will have a status condition and an even higher chance someone will be low on HP. This is when you either switch in Peach or switch gears with Peach and get her to heal up the party with Group Hug. Even if it's only really one party member low on HP, it's still nice to have the other characters healed up. Keep wailing on all enemies with Mario and Geno, and use Maple Syrups when necessary. I liked to use them on Mario's turn, since he deals elemental damage. On the rare turns where Peach doesn't need to heal someone and your FP isn't low, I recommend using Psych Bomb on her turn.
With all the high-power attacks you'll need to perform and recovery items you'll need to pop, I highly advise against using Thought Peek unless there's really nothing else for Mallow to do. And since he should have Star Rain (another non-elemental multi-target move) at this point, there will usually be something better for him to do.
Switch remake-only advice
If you're playing the Switch version, it's important to consider which Triple Move to use. Spare-Us-All may be useful from shielding the party from a particularly nasty attack early on, but it does require you have a full or near-full Action Gauge going into the fight. Healing Rainbow could also be helpful in a pinch if you're out of Finless/Croaka Colas, and it would give Mallow some more utility in the fight. Shooting Star Shot is a strong multi-target attack, and the buffs to both Attack stats and Speed could be helpful as well. I ultimately ended up using Shooting Star Shot to finish off the fight, as Peach had died without any Pick Me Ups in reserve. It was also very cool.
After the fight
Once you defeat Culex, he will have a short conversation with you, then give you the Quartz Charm before leaving. Once he leaves, the door to his room disappears.
Congratulations! You've just defeated Culex! With your Quartz Charm in hand, you can go onwards to Bowser's Keep!
...Or, if you're a Monster List nut like me, to Booster Pass! It's time to use those 15+ Frog Coins to get all the Thought Peeks from the fight! Unfortunately, you do have to talk to the invisible NPC five times over, since they only fill in one entry at a time. But once you do that, then you can go onwards to Bowser's Keep!
Have fun, and hope this helped!