This lady is probably going home after a day at the office. This is a motorbike taxi, you can tell by the orange vest the guy is wearing. A ride from one end of the road to the other (about 500 meter) is 10 baht.
I though I'd add that it is very common for a female passenger to ride sidesaddle, I do it too actually. It is not uncomfortable or difficult at all, and it actually makes getting on and off a lot easier :)
@bkbinthecity - helmets for passengers are optional, a lot of "drivers" also do not bother with a helmet especially if they are only going a short distance or on a small road