As a father of 3 VERY well traveled children I can tell you there are LOTS of things parents can do starting from preparing properly well prior to the flight, to ensuring they have proactively mitigated any potential issued that might arise in-flight. Babies generally cry for a reason, not simply to annoy YOU! The parents should several days before begin to adjust the baby's sleep schedule if that is going to be impacted by the flight. Upon boarding they should make sure they have a comfortable environment set up for themselves and the baby that makes it feel comfortable and secure, particularly if sleeping is hopefully in the cards. One of the most common reasons for a baby to cry on a flight is as a result of the pressure changes on ascent and descent which result in clogged eustachian tubes and painful pressure in the ears. This can be mitigated by having the baby suck on something during take-off...a soother, a bottle, a nipple. Gas is also a leading cause of dis-comfort and subsequent crying, as can be a soiled diaper so the parents can actively manage both of those things as well.

I could go on, but bottom line is there are many things the parents can do both proactively and reactively, and handing the baby to a crew member isn't on the list. If they know or choose to do those things is a different matter, but to suggest there isn't much they can do is wrong.

Now, all that said, YOU have chosen PUBLIC TRANSPORTATION and regardless of your expectations, your Contract of Carriage does not guarantee a good night's sleep or a quiet cabin unfortunately, so I think the likelihood of any compensation is extremely slim...about as slim as the pilot deciding to divert the aircraft to offload the offending baby.