I would say that it is a tiny minority in the U.S. that know what Doctor Who is. It pains me to say this, but mainstream popular culture has no idea. A reference may come up in a Simpsons episode perhaps as almost more of an inside joke, but that is due to the creative guys on the writing team having a broad range of interest, not because a lot of viewers are going to get it.
I VERY rarely can make a reference to Doctor Who without then needing to explain everything from square one. Even my friends who enjoy science fiction don't care about it. They saw Tom Baker's Doctor on PBS and think that it is a kid's. Their loss.
In another current thread a fellow from the UK referred to the very popular and well-known American program "The Today Show" as "obscure". That may well be true for "Today Show"'s notoriety in the UK, but I believe that Doctor Who- unfortunately- may be almost as obscure to viewers in the U.S.
Which is good, to some extent-- as America has a way of taking things from other cultures and transmogrifying them into a U.S. version that audiences here are more comfortable with. I really doubt that I would like that.