The availability of transportation had a dramatic impact on the lives of women, who had traditionally been less mobile.

Is the “who” referring to women or the lives of women?
If you cross out the nonessential prepositional phrase (of women), only the subject (lives) would remain, so isn’t it referring to the “lives”?

No. The “who” here most definitely refers to “women.”

Important point: You need to ignore stuff in prepositional phrases when you’re looking for the subject of a verb, but that is very much NOT the case when you’re looking for the antecedent of a pronoun.