Bringing a child under 2 years old on a plane with you is free for domestic flights if you don’t buy them a seat. However when you travel internationally an infant without a seat needs a ticket, and those can be expensive even when you’re redeeming miles for your own flight.
When you book an award and add a lap infant, some programs charge you 10% of a paid ticket. That’s standard for United, Delta and American. If you redeem miles for a $6000 business class ticket, the lap infant costs you $600 plus taxes.