Preventing a sink strainer from leaking?
Channellocks are great for replacing sink strainer drains. I've never had a sink drain where I needed something like a basin wrench because they are generally easily accessible.

As for the leak:

1)Make sure you don't have a faulty basket strainer. Make sure there are no cracks or other defects in it. You said it looks like condensation underneth so I'm guess that means water is actually on the outside of the plumbing and it's not just leaking past the strainer plug and into the drain.

2) Is it leaking from the sink drain or from one of the drain connections below? Be sure to check the drain connections closely to make sure it's not leaking from them. If you've used putty on top and a rubber gasket and paper gasket on the bottom (make sure the rubber gasket goes directly against the sink and the paper gasket between that and the nut), the likelyhood of it leaking is very slim. You can overtighten the nut (and other plumbing parts that have gaskets) and actually cause leaking issues so be sure not to go crazy on it. Tightness is realitive but it should be tight enough that once you're done you can't twist it by hand and it doesn't move when you tighten the drain pipe back on.  

3) Be sure to check for any chips and/or cracks that water might be able to seep into and leak out. If there are chips and/or cracks, you can try repairing them with some special epoxy or using silicone instead of putty. Silicone will be much more of a pain to remove later but it does come off with some persuasion. If using expoxy, be sure to let it fully cure before putting the strainer back on.