You can’t make a plausible argument that birth control is murder, so it is perfectly possible for a red person to acquiesce to the one, but not the other. If your objective is to grow the domestic population (Douthat) or impose a severe sanction on sexual immorality (most reactionaries), however, birth control is a problem. It needs to go.
Douthat, as a conservative Catholic, undoubtedly opposes birth control and wants it to be banned, but I can’t remember him ever making that case in his NYT column. Why not? He probably thinks it is a bridge too far for his readers. He’s probably right, too; the vast majority of American Catholics ignore Catholic teachings on that issue.
Griswold notwithstanding, I suspect you are going to see some serious efforts in blood red states to ban birth control for the reasons listed above. Let’s hope so, because for the left, it will be the gift that keeps on giving.