Pirate Bases are how you survive in the beginning of the game, but later on they become a detriment to your progress. The way it works is (assuming you're playing standard pirate with standard base costs) you pay 30,000cr and build a base on a world. Once finished, you will siphon of 10% of it's GDP, just as if you were taxing it, even though you don't actually own the planet. Add a pirate fortress for 100,000cr and this gets raised to 30%. A 30% tax rate is on par with sane tax rates as an empire, though they don't have to dump 130,000cr investment to get there.
Empires will always try to destroy your bases, however, therefore it only practical to place pirate bases on independent worlds, because they don't fight back. You can't see the GDP of an independent world, but you'll still get your cut of it. What is worth noting is that there is a cut-off on building influence. Around 1.5B population merely having ships in orbit will no longer build pirate influence. Past around 3.5B or so even raiding no longer gives you any influence. A planet with a population over 4B is effectively immune to pirate influence entirely. This is worth keeping in mind because after enough time even independent worlds will gain too much population for you to influence. Therefore you need to either act quickly or be ready to rain nukes on planets to bring their population back down to where you can influence them again.
Since empires will try to destroy your bases the only logical reason to put pirate facilities on an empire controlled world would be when you are planning to take it over outright via the Criminal Network, which will run you another 200,000cr on top of everything else. So you're looking at having to earn a positive balance of 330,000cr, after expenses, before your target gets too big to influence. This is not easy and you'll be very grateful that it only needs to be done once. Ideally it should be done to the homeworld of whatever race matches that of your pirate faction. Thus if you are Human pirates, then you want to take over the Human homeworld with a criminal network.
Once this is done the planet is now officially yours, and you should scrap both the network and your pirate bases. Why? Because now that 30%+ "corruption" tax is no longer going into your wallet. Instead it's just bleeding it out into the nearest black hole. That's right, they're no longer earning you money - they're costing it. I'm not sure if this is a bug or not, but since development support for this game has long since ended, it's just how it works and there is no sense in protesting the sheer illogic of it all. Keep that in mind for any other planets you properly conquer later. Once you actually own a planet your pirate bases are a liability.
However, getting a homeworld of your own is essential for moving forward and should be the number one priority of every pirate until that goal is achieved. You can continue placing pirate bases on independent worlds afterward, but that first planet taken over is most important task you have, because doing so ensures victory and failing to do so ensures defeat. Plus having pirate bases creates such a massive boost to the "ceiling" of your research points that you probably will never actually hit that ceiling no matter how hard you try.
Also, you can build cheap defense platforms over independent worlds and I highly suggest doing exactly that. First, it lets you take any defense mission contracts that they post completely risk free. Second, it usually keeps empires from colonizing it (however in large games AI colony ships will magically slip past when you're not looking). Third, it keeps rival pirates from screwing with it. You can also build cheap defense platforms over empire homeworlds and I suggest doing that even more. Once you've destroyed their homeworld spaceport you should immediately send your construction ship there (with escort) and build a defense base on top of it. Doing this will effectively take that empire out of the game entirely. From that point forward any time they try to build a space port or construction ship, it won't get past the first build cycle before your defense base blows it away. You may not have enough army troops to conquer them just yet, but so what? They're not going anywhere. Take your time.