What running shoe provides the best arch support for an active person
with flat feet?

I have the opposite problem: high arches. But a friend of mine who has
flat feet is very enthusiastic about Brooks Beast running shoes. They
aren't cheap, but they are great for flat-footed runners.

http://www.epinions.com/content_16047705732

http://www.shoebuy.com/sb/s.jsp?Style=61748

I have natural flat feet. I have searched all my life for arch support
quality shoes. Almost all brands have a arch insert of some type.
However, they are general in arch height. Which translate to improper
fit, which causes pain if not a custom fit.
There are many foot types and 3 important measurements that are needed
to obtain a useful product.
Archatomics-custom arch supports at Archatomics.com will explain some
typical foot problems, as well as offer low-cost($68) custom fit arch
supports. It is all done online and is very simple. I have been
wearing mine for 4 years and you can move them in and out of all/any
of your shoes.
They provide proper support and balance, which equivelates to lower
body pain relief.
And they last 5-10 years.
My advice is check out Archatomics.com
You'll be happy that you did.