Just to add to the debate, my understanding is that these borders are usually to do with the tool that it used to crop, and there being some feathering going on, as was suggested above - e.g. using a marquee tool with the feathering at anything other than zero. You shouldn't get this effect with the crop tool itself.
I would also concur with the others above that you should do your editing on your resized image - i.e. cropping/resizing should be in the first few steps of your editing workflow. Though this is less crucial for things like brightness and colour adjustment, sharpening different sized images using the same sharpening settings will produce different results. For example, getting your photo just right in terms of sharpening and then reducing the size of it will have the effect of exaggerating the sharpness (as you now have less pixels that make the photo) and so you are more likely to get jaggies.
Apologies if I am preaching to the converted here, but I think this factor does make a difference to the final quality of the image.
All the best.