This happened in my game earlier today. I agree; it was absolutely devastating... to the AI's militaries.
The Arceans were unable to build any ships at all for fear of being blasted out of the sky by pirates. Although I had intended on taking the Arceans out after the more worrisome Korath, I changed my mind when I saw that the pirates were doing a lot of my pre-invasion work for me. A few fast transport ships to slip by the pirates and the Arcean empire was absorbed into my own in no time.
I had the advantage of being fairly high up in military tech, which I had been to protect myself from the Korath. Huge ships with high armour (or maybe point defense) were able to take on fleets of pirates and gain valuable experience from it. They did manage, however, to take out my active fleet of medium sized ships and most of my starbases before I was able to get the first huge ship out of the stardocks.
I supppse the pirates could have been more devastating had I not been prepared for them; at least they were an equal opportunity disaster!
In your situation, focusing on influence, you need to worry about quick AI transports hitting your worlds. But since you have high populations, presumably, that may not be a big problem. The AI ships tend to be slower and they have to get through pirates protecting your planets. You might find focusing on Soldiering techs helpful, just in case. Meanwhile, let the AIs deal with the pirate problem, if they can. If they can't, you're no worse off, at least until you can build a ship that can deal with them yourself.
