Part of the problem is that saying "anime" is like saying "tv". If I show you a random series like Pokemon or Madoka or TTGL, you've learned almost nothing about "anime", only those three series.
Specific series with concepts and topics that already appeal to mainstream WOULD probably do well.
Toradora and Love Complex are pretty classic rom-coms ala Cary Grant/Audrey Hepburn. They've done well when I've shown them to non-anime people.
Strawberry Marshmallow, several non-anime women thought the series was darling.
K-On! is a coming-of-age and bonding story, a regular afterschool special.
As for the "cartoon" problem - that's more of a problem for older people and that portion of society that don't get much of anything - the "muggles". Those people who work in a pointless job, go home and eat pre-processed food, then watch American Idol and Inside Edition sitting passively on their couch. They usually make fun of anyone who does something different.
Screw them.
