I went up White Mountain last Wednesday (June 12). The dirt road was actually not too bad - I've certainly driven on far worse. I had a 4WD Subaru Outback, but I still drove no more than 30 mph because I didn't want to destroy my car. We saw a Civic and a Camry parked at the gate, so I'd say it's no problem getting up there with a regular car if you're willing to drive a bit slowly.
