It seems to be a very complex question. Or better said: the answer depends upon what is "culture".
My first thought is that culture changes (I'd call these culture downgrades) are symptoms of unfaithfulness. In other words, judgments from God. If that is true, then the changes in culture are not causes that hinder the gospel (because they cannot) but are obstacles placed before us by God for our chastening and humbling.
That's one reason I don't go on cultural crusades per se. It's like fighting an abstraction.
But, having said that, I am passionate to promote the gospel and to argue for justice, whether it be political, judicial, economic, or otherwise. That is grounded on what I understand to be the mandate from creation: subdue the earth. We can only do that properly by obeying our Lord's commandments.
So, I personally don't set out to redeem culture, but rather focus on what I see is wrong in my own sphere, bring it to light, and demand change of the particular practice (not the abstract wrong). In calling something wrong, the only standard I can legitimately rely upon is scripture. Of course, we all say this. It is hard for us to really do this.
In other words, culture is too big for me to understand. I don't have the knowledge, skill, or capacity to figure out how to change it directly. But I can decry the evil culture in its specifics. As in the case of murdering the unborn, I think it is more in my grasp to thoroughly and broadly convince people that this practice is horribly wrong, call for repentence, and let changed hearts change culture. So it goes for all our woes. I don't think we will ever succeed in changing culture by attacking it directly.