I have 2 secrets for bright, beautiful pesto:

1) Add just a tiny bit of fresh lemon juice. I use 1 tablespoon per 12 lbs of basil. The citric acid acts as a preservative, and the almost negligible amount of added acidity gives the pesto a nice brightness in flavor.

2) Blanch your pesto (just for a second!). Really, just for a second. I put my stemmed basil in a colander and pour boiling water through it, then dump in some ice and run cold water over it. If you blanch it any longer you'll lose a lot of flavor, but I found that this quick blanching knocks back the grassy flavor while retaining all the great herbal notes.

This is the only way I've found to have bright, beautiful pesto without losing any flavor. I've tested it in my fridge and after 2-3 weeks it's still as bright as the day I made it.

You can pour a little olive oil over your pesto if storing in a plastic tub.

I read recently (don't recall where) that it's the water on the basil leaves that is the culprit, and so I made certain that the leaves were totally dry prior to making my next batch of pesto in the food processor. It worked - bright green pesto!

You can blanch and dry your basil leaves before processing. Your pesto will stay a bright, lovely green.

Basil tends to oxidize if it's over-pureed. The food processor does this more quickly than the blender. I usually make pesto in my blender to avoid oxidation.