It really depends upon what kind of anti-gunner it is with whom I'm speaking.
A. If it's one that thinks he has a rational / utilitarian argument, I'll rationally pick apart his argument with logic. (1) There are too many utilitarian facts that favor private gun ownership such as the inability of police to be everywhere and the inverse correlation between private gun ownership rates and crime rates. (2) I'd detail the flaws in their arguments and the pseudo-science fallacies that are espoused by typical anti-gunners. (3) I'd also point out that, Constitutionally speaking, guaranteed rights may be in direct opposition to what is favored by utilitarianism but are guaranteed nonetheless -- that the guarantees of certain rights that we enjoy are often in spite of (not because of) their level of popularity.
If they are truly grounded in rational thinking, their position is weakened and some hope is born.
B. If it's one that's anti-gun because he's never had reason to think otherwise, I'll simply invite him to the range. Very often, these people have an anti-gun sentiment based upon unfamiliarity and fear. I'd even spotlight their fear; if a guy breaks into your house with a gun, you're going to be terrified but if a cop shows up with the very same kind of gun, you're going to be relieved. So it can't be the gun that's the problem. Typically, if they accept my invitation and come to the range, they'll see the cameraderie, the families, the fun, etc. and their position starts to soften.
C. If it's an anti-gunner that's firmly entrenched for whatever reason (e.g. Schumer, Feinstein), I don't even bother. I've spent enough energy trying with people like this to have come to the realization that I'm just spinning my wheels. Despite the logic of approach A or the enjoyment of approach B, these people refuse to budge. That kind of dogmatic thinking is near impossible to change. My time is better spent getting people who are closer to the fence to swing over to our side.