That's a challenging topic. I suppose there are several angles of approach, depending on the person you need to convince.
My wife and I have used NFP (Natural Family Planning) our entire marriage (15+ years), to great success. Maybe that's the first argument - there are "natural" ways to avoid pregnancy that work very well. So contraception is only necessary if the couple isn't willing to try NFP.
There are health reasons - the "pill" has many negative side effects for women. Most woman (and men) probably have no idea, but some simple research will scare you. Most "medicine" would be pulled from the market if it had those side effects. By definition it's "un-natural", at a time when many people are concerned about moving closer to nature.
Of course, there's the theological argument, which gets much deeper. If your spouse is so inclined, there are many good resources on this. One question might be "Do you understand why the Church has this stance?" If not, then why not make the effort to understand, before dismissing it.
Remember also to pray - God works miracles over time.