Another data point: My wife's grandparents have 10 children, 81 grandchildren (mostly adults), and around 300 and counting great-grandchildren. It's wonderful! When we go on trips around the U.S., we are almost always staying with a cousin. Our children really enjoy meeting and getting to know cousins their age. That said, I do notice that the younger generations typically are having smaller families (like 3-6 children instead of 8-12 children).
It's a lot of work to keep in touch and visit cousins, but my wife and I feel it's totally worth it. At times, we've decided where to live based on proximity to cousins, and we decided early on to regularly visit cousins who lived within a couple of hours. When we were married, we decided we'd try to go to reunions, funerals, and weddings, and we've largely done that for the last 19 years. There is a big family reunion for a few days every other year, with about half the family coming (so hundreds of people of all ages). We also have monthly dinners with typically 20-30 local cousins joining. We regularly tell family history stories, and we have a screensaver rotating through our pictures, including many from cousin visits and reunions, so cousins are regularly coming up in daily conversations.
One thing I've noticed is that the more often we go to reunions and visit other families, the more excited the children are to go to see cousins again (as opposed to another family branch, where reunions are every 5-10 years, the children just don't have the same connection with that much growing up in between reunions). Another thing I've noticed is that our children seem to feel very grounded in their experiences - they often are relating their experiences to a story from an ancestor and pulling lessons from that.
To address a comment elsewhere on this topic, yes, we are religious, the family is predominantly active members of The Church of Jesus Christ of Latter-day Saints, and a significant portion live in the intermountain west area of the U.S.