So far the Obama campaign’s respose has been that Obama ran as a Democrat in that campaign.
Of course he did. The New Party was a fusion party that generally ran its candidates as Democrats. Also important is that Obama had to sign a contract with them in order to get their endorsement.
Politico did look into Obama’s 1996 campaign. They posted this answer he provided to a questionaire which says he’s running as a Democrat and admits he was “endorsed” by the New Party:
http://www.politico.com/static/PPM41_obamaquestionaire1newest.html
If you look at that questionaire, it’s really left-wing stuff. No decent politician would even brag about being “endorsed” by the New Party, but I’m sure Obama will use the questionaire as proof that he wasn’t an actual member of the party.