Regarding the Hidden Pirate Bases and Hidden Pirate Fortresses: go to the Colonies window, select the facilities tab, select a planet with the pirate base on the world, select the pirate base, click the "attack" button. I would strongly suggest that you have at least 5 units of infantry on the planet before doing this, and if your infantry units are of a relatively weak species you may need something closer to 10.
I don’t have the economy to pay them and I don’t really have enough military ships to protect all my assets.
Pay the pirates long enough to establish a gas mine and a fleet presence in the area. Don't try to protect everything, and try not to build mines in outlying systems. Instead, preferentially build mines within colonized systems with spaceports in them, particularly if there's a fuel source in that system, as even with Warp Bubble Generators you can then use one fleet to defend everything in the system (3 or more warships of at least 200 size should be adequate, though it depends on how strong and how numerous your pirates are), and your fleet will have easy access to a center for repairs and fuel. Keep 5 or so units of infantry on most colonies, as this will adequately defend against minor raiding. Try to keep new colonies fairly close to established colonies so that you can use the established infrastructure to support the defensive forces; for more distant colonies, then you'll probably want to develop and deploy a resupply ship to act as a temporary refueling point and send a larger fleet out that way (or have rotating fleet assignments, so that at least one fleet is coming on station while the previous fleet is headed home for repairs). Note that 'distant' varies depending on the quality of your hyperdrives and fuel cells, and the size of your ships - Warp Bubble Generators are so slow and fuel-hungry that 'distant' could be as little as half a sector away, especially if you're still on smaller ship sizes. Also make certain to grab nearby independent colonies, or at the very least drive the pirates away from the independent colonies, as these are an incredible source of income and fuel for the pirate factions and are only rarely large enough to cause the pirates difficulty for constructing hidden bases and fortresses.
You should also start preparing a large fleet or two to go around busting pirate starports and construction ships. Stealing territory maps is usually sufficient to locate the starports and controlled independent colonies, but you'll need operations maps or deep cover or long-range scanners to effectively hunt down and kill construction ships and mines (lots of exploration ships is also workable, but it's less effective than long-range scanners, particularly since the explorers tend not to return to already-mapped systems). Long-range weapons are good for base-busting fleets, as an individual ship is unlikely to be able to survive a close-quarters fight with a station, so missiles, torpedoes, and fighters are good priorities for weapons research. Focus on killing off the closest pirate spaceports first, as this will degrade the various pirate factions' ability to attack you. Remember that pirate bases usually crop up at sources of a fuel gas, so keep an eye out in those areas for pirate construction ships, pirate gas mining stations, and pirate resupply ships (this can be done by building your own gas mining stations in these areas, though this isn't always advisable, especially if you haven't heavily fortified your mining stations). Also remember that pirates, for the most part, will only be able to refuel at their own bases or at independent colonies, and this is an important reason to incorporate the independents into your own empire as soon as you can. Denying the pirates the ability to refuel nearby goes a long way towards driving them out of your space.
You can also design your mining stations to be heavily fortified. If you do this, you'll want to build the base up to the maximum allowed size for construction away from colonies, you'll want to go with fairly heavy shielding, and you'll want tractor beams to keep the pirate ships in close to the base combined with a decent close-in armament. It is not a bad idea to also keep a decent battery of missiles or torpedoes on the station, but the pirates will likely close in to blaster range before the station really starts fighting back, so a tractor beam or two and a good close-in armament is a good choice for killing off pirates that try to attack the station. Another alternative is to use graviton beams and hope to drive the pirate off due to component damage before the pirates can do too much harm to your station's shields, while adding boarding pods to your station's design will make it more difficult to successfully raid or capture the station and can turn pirate raiders into new ships for the fleet (or for decommissioning for technology or money).