Actually, it depends on the version of PSP 3000 as some of the newer ones need modules in the newer firmware. You would need to use something like pspident to find out which one you have (5g, 6g, etc). I think the downgrader will just refuse to downgrade an incompatible 3000 but I am not sure so don't rely on that as if it does try to downgrade you could brick.
1000, 2000, 3000, they're not single models. They're more like classes, and the actual motherboard models vary. This also means that the base firmware they came with varies. You can buy a brand-new PSP 3000 today and it will most likely come with 6.60 preinstalled. You cannot naturally downgrade a PSP beyond what it came with.
Now, while there is a method to downgrade a later 3K back to 6.20, it can have complications so I don't recommend it since sleep mode doesn't undo the hack on 6.60 anyways.
I have a 3000 model 4g and downgraded successfully from 6.60 OFW to 6.20 OFW then installed the 6.20 PRO perma patch. The only thing I lost after downgrading was my XMB settings. The downgrader that I used checks your psp model before you initiate the downgrade. I also used psp module checker to check as well.
supposedly if you have a 7g or 9g model you can't downgrade.
the downgrader I used can be found here (Although it worked for me, I'm not sure how reliable it is. Make sure you read the forum posts.)
A lot of people caution against downgrading and using the 6.20 PRO perma patch though. If you can live with the 6.60 PRO fast recovery then that would be a safer route. If you do go the perma patch route, make sure you back up the vshorig.prx file it saves to the root of your memory stick.
If you have a 7g or 9g a downgrade is technically possibly by forging some information, but it has possible drawbacks (some people reported being unable to upgrade again afterwords, for example), and seems riskier than other modern procedures so I don't recommend it.