I don't have kids but all my friends do. I've never had an interest in kids and never wanted any of my own.
For friends I take a mild interest in them, just a basic how are they doing. If they are with mum then I'll interact with them otherwise that's about it. I don't buy them gifts or go to parties etc. I wouldn't be comfortable babysitting them as they don't know me well. But in an emergency I would definitely help out if no one else was available.
I've babysat for a couple of really good friends but this has always been for special occasions and not something I would do regularly. But I'm more like aunty to these kids and I do see them much more regularly, I will do small gifts for Xmas and birthdays etc.
My nephew is a bit different, I see him much more often and have had him for sleepovers. We've even taken him for weekend breaks etc. I'm the cool aunt with the sporty car and like to treat him. But I do come from a close family and we do have weekly meals with parents. Sister was widowed when pregnant, at the time I worked shifts and had lots of free time. I guess I was like a second mum to him, I helped out lots during her pregnancy and the first few years of his life. So I am really close to him. Probably wouldn't be the case if she hadn't been widowed. Her other son I'm not as close with, yeah i still spend time with him etc but he has both parents too so my sister doesn't need support.
It depends on the parents and how close I am to them. The friends I just ask in passing I don't see often anymore (all grown up with our own busy lives) but still have the occasional night out/catch up. The good friends I've babysit for I see at least weekly.
Neither my sister or my close friends have ever asked me to babysit - I have always offered. I think the new mum seems like she's expecting lots but it should be down to you what you want to do and how involved you want to be. I would say set boundaries though from the start