If you have 1000+ comments awaiting moderation, the only possible explanation is that, most of them are spam. Because, if your blog is that popular, you would know what to do already and wouldn’t be reading this at all.
What you need is Akismet. But you probably heard about it already, installed and configured it, and no, nothing happened.Its filtering the spam after its install, but what will I do with the ones I have already?
You are missing the magic button. On the Comments page in the WordPress admin, you will see a button, Check for Spam:
Click it. Depending on the queue length, it will take a while. I just did this for a website, they had around 2500 comments awaiting moderation and it took about 2 minutes.
Akismet says: 83% of all comments are spam 😉