This is how you win over a mildly anti-gun person: with calm, rational discussion and a personal example of being reasonable and level headed. Let them see that you are not sitting there dreaming of shooting people.
We graduated to firing a suppressed 9mm M&P, then the suppressor came off, then we worked up to shooting a .45. She gradually accepted the fact that I carry a weapon. At first, it was just when we went to Orlando, the dangerous place. As she grew to accept it, I carried more and more.
The shooting in Paris actually was a turning point. We were on vacation in France last week. When we saw the news of the shooting, she turned to me and said, "When we get home, I want to take the concealed weapons safety course, so I can apply for a concealed weapons permit."
That is how you turn a mildly anti-gun girl from New York into a CCW holder:
1 Antis from places like NYC, Maryland, and Massachusetts are taught that gun owners are all a bunch of mouth breathing, uneducated, racist hicks who sit around and dream of killing minorities. Don't be that. Be well spoken, level headed, and non-violent.
2 Calmly approach the subject, and get them to shoot something mild and quiet. Nothing will reinforce the "guns are scary" meme like trying to get a female who is afraid of guns to shoot a .500 Magnum while you yuck it up as the gun recoils into her face.
3 Don't pressure them. They will eventually see things for what they are. Once they begin to see the world as it is, not as their indoctrination said it is, they will come around. Remember: facts and reality are on your side.