It depends. On roads where there is traffic it is a petach patuach lareshut harabbim. Giving a ride to a married woman might be easier if her husband is in town. Someone who wants to be machmir can have his cell phone film the inside of the car (I know someone who does this to protect himself from blackmail), In some situations it might be a mitzva as the woman could be in danger if she stays outside by herself.
Yungerman, please cite Rav Moshe’s teshuva so we can see what was the exact situation he discussed.
Eli, once a man offered me a ride. I thought there was something on the front passenger seat so I sat in the back. He was very offended that I made him my chauffeur. I was also chided by one man for not closing the door enough after getting out and by another for slamming it and possibly causing damage. At the risk of going off topic, I will also add that a passenger, no matter who is the driver, should act with derech eretz. Do not eat or conduct cell phone conversations without permission (they annoy some drivers). Thank the driver and give him/her a beracha (e.g. “kol tuv”, “shavua tov”).