You run a business, they run a business, and everyone is trying to make money. How much money do you think an IE6 user has? the browser is like a decade old. It's why we target iPhone and not Android in the mobile world. ( because iPhone users out spend android users. )
Tell your client the truth, tell them that you keep up to date on the latest trends and if they want to use someone else let them, and then tell them exactly what your work will do.
I would not even offer to support IE6 anymore. Maybe IE7. There are a few reasons.
#1 you can't download IE6 legally any more
#2 you can't install it legally on linux or mac systems to test.
#3 IE7 in quirks mode does not work exactly the same as IE6.
#4 web development for legacy browsers is less about building things, and more about working around problems. and i just don't find that fun.
It is your duty as a web professional to turn down this work. Microsoft does not even support his browser anymore.