Is there an idiomatic word or phrase for the youngest child of a family? It may somewhat have a negative connotation since their parents usually infantilize them.
Your use of "infantilize" provided a clue: the baby of the family.
From the article "Youngest Child Problems: 9 Things Only The Baby Of The Family Gets" by Margaret O'Brien:
Being the baby of the family isn’t always as glamorous as many people think.
For years, oldest and middle children have been claiming that the youngest “gets away with everything” and is “clearly the favorite.”
They have a sort of prejudice against the babies of the family, and it's time the issue was addressed. What they don’t realize is that being the last born comes with its own set of problems.
You've used one of the best ways to say it in your question: the youngest.
Although this is often stretched out to say, "the youngest in the family," or, "the youngest member of my family," it's also sometime shortened to "the youngest," particularly when the context makes it clear you are talking about a person and their siblings. Another common way to say it is, "the youngest of X," where X is the number of children in the family.
Here are a few examples:
Most people say the youngest child has it the easiest. They’re the spoiled ones who can’t do anything wrong, after all. But if you’re the youngest, like me, you’ll know that’s not true.
There were four children in my family. I was the youngest and the only girl.
I was the youngest of four, and the only girl.
He was the youngest of the five sons of Jennie (Gorovitz) and Solomon Horwitz. Because he was the youngest, his brothers called him "Babe" to tease him.
He was the youngest of four, and his parents took him to see musicians performing in the area.
Lucia, her parents and three brothers were all recruited into the business. There was no escaping it. She was the youngest but she became the face of the restaurant.