幼子 sounds a little old, and we don't use this word in daily conversation. (not so old as we can't understand the meaning.) It also sounds formal, so it is sometimes used in essays or titles of books.
Instead of 幼子, we often use 幼児(youji, same as 幼子) or 乳児(nyuuji, a breast-fed baby). 赤ちゃん sounds They sound more casualformal than 赤ん坊 or 赤ちゃん.