I used to have terrible approach anxiety. I couldn't even say hi to a stranger in the street. Now I have no trouble starting conversations. It took me 3 weeks to change.
Short answer: practice.
Long answer:
1. Say hi to everyone you walk past, nice and loud, with a smile.
Set yourself a goal to say hi to at least 20 people a day, every day, for a week. Say this to men, women, old, young, etc.
At first it will be hard, but it will get so easy by the end of the week that it will feel weird not to say hi. After a day or so, you should have the courage to say hi to two people walking together. Remember to do it loud and with a smile.
2. Progress to conversations with women. The opinion opener is the best if you have approach anxiety.
Ask if you can have their opinion on something. It doesn't matter what it is, so long as it sounds sincere. I've never had a woman decline to give me an "opinion." Go out and do this to at least 5 women every day for a week. If they seem interested, ask their name, and go from there.
Keep your body language non-needy. Lean back, not forwards. Don't directly face her while talking until she has earned your interest. If she seems interested, don't stay on the opinion opener for too long, you need to start properly gaming her.
The opinion opener is just that, an opener. It gets your foot in the door. You have to go from there.
I could say a lot more about approaching and how to beat approach anxiety, I've been thinking about doing a post in the "Don Juan Tips" sometime soon. Keep an eye out.