I don't know if it makes a difference but are you using purple basil? I really don't know why it would turn brown otherwise... are you putting the pesto in at the last moment? If it's overcooked maybe that might be the reason? I am sure there are other people who may offer a more professional explanation than me.

Its something of a hassle, but you could blanch and then shock the basil before bashing/chopping it up.  That will  set the color for literally a week.

Most likely the combination is 2 things:
     1: The pesto is made slightly too far ahead, allowing time for the chlorophyll to oxidize and turn brown
     2: The basil is exposed to too much heat.  The best way to dress with pesto is over no heat, just drain the pasta into a bowl, put the pesto on top and toss

I hope one of these helps.  Try making the pesto closer to the finish first, as its less of a mess and a hassle than shocking the basil would be.  However, if that doesn't fix your color issue, shock away, it will be greener than the Sierra Club for days on end

WTM

I agree, you'll find your pesto(basil) is oxidizing.
And you don't cook pesto at all. It's carefully mixed into the pasta just before serving.

You cook pesto with chicken???

I'm sorry, I'm a traditionalist.
Pesto is for pasta.
But I suppose we all like different things and there's nothing wrong with experimenting.