Hello S.,
It is very simple, really. It will a bit tough, but you gotta do it.
I would make an appointment with her first (the baby room worker) and just express how you feel about when you are dropping off your baby. First of all that takes up alot of your time and you only have 5 or 10 minutes to give her daily instructions and need to go. I would just be direct, but firm (not mean) and just tell her that you would love to be able to chat, but you just don't have the time.
If it continues, then you need to speak with her supervisor. This is totally inappropriate behavior and simply not acceptable subject matter to be discussing w/parents of the daycare that is being run. You are not her psychiatrist or her counselor/pastor and you really don't have the "background" to offer any help w/her depression or whatever else she is going through. (Again, play sympathetic, but be firm) Please follow the chain of command, speak w/her first, then her supervisor and the school director if things are not being handled. Don't wait long in between to see an improvement, do it quickly.
The Director of the school needs to be notified about this. Each teacher needs to represent themselves in a positive manner whether their day is going their way or not. Each teacher represents that school and that Director. That is why you put your child in their care and you don't want to hear tragic stories and life-altering moments each time you drop off your child.
Your need to know for your own peace of mind that your child is promised a fun and positive day!
Good Luck!
G. B.