Can Curly Brace survive?

Yes there is, though the method to do so is rather arcane.

The first "make or break" moment has nothing to do with Curly herself - when you see Dr. Booster fall into the hole Labyrinth, you must not follow him. Following him at this point will earn you the Booster v.08, and Dr. Booster will also die.

If you do not follow him, Dr. Booster will Survive, and, much later, provide you with the Booster v2.0

If you have done this, you will be able to pick up the "Tow Rope" item, and pull Curly Brace along behind you when you flee the area after defeating The Core. Make sure to pick this up before starting the boss fight (it's in one of the bottom corners).

Once you're in the waterway, make your way to the waterside cavern, put Curly on the bed, use the computer to figure out how to drain the water out of her, do so, and then take her with you again when you leave.

When you destroy the fish boss during the waterway fleeing segment, Curly will automatically be dragged away from you.

If you manage to do all this, Curly survives, and you can find her in the lower right area of the plantation, though she'll have lost her memory. To fix this, you need to go to the Village Graveyard (you'll also get the Booster v2.0 here) and use your new booster to get to a mushroom you need to restore her memory.

Good luck!

Yes, there's a way for she to survive, and it's necessary for the best ending of the game.

When you're at the labirynth, and you enter a room where dr booster falls, you have to ignore him, so it's best for you to save your game before you go down at him (you can't fall on the hole to talk with him), and just jump to the other side of the room and enter the door, this way you won't get the booster v0.8 (but now you'll be able to get the v2.0 later).

Then on the room where you fight The Core for the first time, on the bottom right corner of the room, you'll find the Tow Rope, grab it (you can do it before or after the battle, but you can't leave the room because a block won't let you return), and after the fight you can grab curly and take her with you.

At the Waterway, you enter on the only house, take a nap, and check the pc and the bookshelf until you learn how to remove the water from robots, then take off curly's water and check again the pc until he says to you good luck. the take curly again and keep going normaly, when you defeat the fish boss (tip: if you manage to defeat him without taking damage you'll win a secret useless item xD) she'll disappear.

You find her again at the Plantation without her memory and you'll have to fight the Ma pignon on the secret room at the cemetery from Mimiga Village (here you'll already have the booster 2.0; there'll be a little quiz, nothing hard), and when you give it to her, she'll remember everything. and this way you'll go for the best ending (but there's a lot to do before the end.)