Yes you can turn them off, before you start the game, there should be something that says game options where you can pick how fast the game plays, if teams are locked, fleet sizes and turning on and off pirates.  You can't turn them off once the game is already going unless you consider destroying their base as "turning them off".

When you host (SP or MP) a game (i.e. setup player slots, AIs, teams etc..., there is an Options button at the bottom. Click on it and it opens a window that allows you to turn pirates on or off, as well as a number of other options for the game you are about to host. Others can see the options but cannot change them as it is a host setting...

I have no options button in the aforementioned locations. I am using v1.00.

To add to what Annatar said:

Getting the updates to the game will really help you out with a bunch of improvements.  Graphical and the User Interface are among those worth mentioning, but it's almost always balance issues that are fixed.  Currently you can just update it by using the Impulse Client.  If you are among the 4% of people that are having trouble with it (Hint, have your firewall allow Impulse ), then websites like Filefront have the 1.05 update to apply it.  Most people at this point already upgraded, so unless you're hosting a LAN, good luck trying to find some games.  I would go ahead and update now, because each update makes your save files inert, making you start new games.  It's a bit off topic about pirates, but I support Ironclad for supplying updates worth a dang.

Now that I think about it, I can contribute to the pirates convo.  It took me like 3 hours to figure this out, don't share my shame .  There are some maps (and those you can make on your own too) that will not have pirate bases.  Even if you have pirates enabled, they will not show.  Quick tidbit for you while you're still in the learning curve