Maybe it’s because I’m a guy, but I would focus on the “logic” aspect, where sex is designed for two purposes and to purposely thwart either purpose is the problem. The Church isn’t against artificial contraception for some arbitrary reason, such as wanting lots of kids to fill the pews. It’s to avoid twisting nature, and thus twisting the true expression of love.
OK this is going to sound very heartless and legalistic but for guys, surritter’s approach will work very well and I would go even further than that, I would talk about God the sovereign Creator and Master of the Universe, that this is how He created and wants to run the universe, that all the Early Church Fathers condemned contraception and sterilization in the strongest terms, and that we should not be kidding ourselves that it has suddenly become acceptable to contracept, just because “everybody else is doing it”, just because all Christian Churches and denominations (except the Catholic) have suddenly abandoned their former teachings against contraception after the Church of England’s Lambeth Conference of 1930.
So, I would take - with the guys - a heartless and legalistic approach

and throw in, for good measure, some 4-letter words

by which I mean, of course, H-E-L-L and N-O-A-H (as in Noah and the flood). I would give them a straigth explanation, without sugar coating, that contraception is a mortal sin that can condemn one’s soul to eternal damnation in the H-E-L-L, and, as soon as someone would bring up the argument that “but everybody else is doing it”, I would mention that yeah, everybody else was doing all kinds of bad stuff in the time of N-O-A-H, and they all perished in the flood except for Noah and his family, because God will not put up with mortal sins, even if seemingly “everybody is doing it”.
Don’t worry, this approach is quite effective and works like a charm with guys.

I’m not going to tell you how to convince ladies, but I’m a man and I know how men think. They will appreciate if you will give them some straigth talk with the facts. You may also want to throw in some quotes from the Early Church Fathers and even the example of Onan from the Old Testament just to demonstrate that contraception was never acceptable throughout the Old and New Testaments, that we Catholics have been faithful to the constant teaching of the Church established by Christ throughout these 2000 years, and that the current acceptance of contraception in the secular Western world and even by Protestant denominations (and Eastern Orthodox Churches) is nothing but a recent heresy, a heresy that started in 1930.