Not a drunk driver, but teenage boys who terrorized me by speeding past and playing "chicken" while they were on electric 4-wheelers. Resulted in a concussion and a large gash on my leg that required 20 stitches. I was 12 then.
About a year ago I decided that I wanted to learn to ride a bike again. Here are the most important things that helped me:
Off road - NO traffic. I practiced in an empty basketball court until I became confident enough to go on a paved trail with no one else on it.
NO KIDS! I never could have done it with my kids along. Too worried about what they were doing and whether one of them would come too close to me.
Small bike - I'm 5'0 and I started out on my child's 20" bike with the seat set up higher. The fact that my feet could touch the ground (flat-footed) at any time was VERY reassuring. I'm on a bike that fits me now.
The big thing is CONFIDENCE. I had NO confidence in myself or anyone else who was riding near me (or even walking near me). It took awhile to build it up - be sure to reassure her A LOT!
As for your approach - be honest. Tell her that you understand her fears and JUSTIFY them! Tell her she has a right to be afraid! Tell her that you want to make it a family affair - that you enjoy her company and so do the kids and this is an experience you want to share with her!
Of all the above points, I think the most important one is to take her out to ride without the kids. It might even be best if YOU are not on a bike either - that way she feels a little more powerful and doesn't have to worry about how close your bike is to hers or where you are.
So now *my* challenge is to get my husband interested in biking!! I've used the honest approach, but he's not at all interested.