No. It's one thing to set up a pirates den with brothels, gambling and bars. Quite another to set up and run successfully a harbour with extended port facilities. Find a remote spot. Build some shacks. Import some women, playing cards and drinkies. You're all set to go.
Building a harbour or port is of a different order. Adding a dry dock to that even more. Most ports and nearly all harbours didn't have dry dock facilities. Only the largest ports had them. Because they are expensive to build, maintain and operate.
That brings us to profitability. The Caribbean Sea may have been a pirate's paradise, over all there weren't that many pirate ships. A pirate dry dock would run at a severe loss.
Next, there was no real need for dry docking pirate ships. Pirates never used big ships, like galleons. They preferred small, nimble and fast ships. It's not a coincidence Somali pirates do exactly the same thing.
There are three reasons why pirates wouldn't need to use dry docks:
1- You don't need to dry dock a sloop or another small type of fast ship. You can simply beach it, and work from there.
2- Steal another ship.
3- A dry dock would be a magnet for all European navies. So many pirate ships nicely docked waiting (defenceless!) to be serviced in one harbour is a wet dream for an admiral. That's how he would get rich in a hurry. Absolutely no need to motivate his crew. They'd be more than happy to do their very best.