karadaliev: I can confirm this as well - you need at least one destroyer over the pirate planet and pick the attack option (and nuke next).
You have at least 1 nuke on each destroyer per turn, so if you didn't manage to destroy the base, you can try the next turn, or the next turn, and until you manage to destroy it.
Keep in mind that the pirate ships just randomly spawn over a pirate planet, so don't leave a single destroyer there, protect him with other ships until the base is destroyed.
BTW today I saw the pirates generate a base on the same planet 3 times.
Yea. I destroyed a pirate base on some asteroids, wiping out the everyone. Left it. A couple of turns later, pirates were there again. It just seemed a little quick for pirates to repopulate the area so quick.