I know a nice daycare center in the Bronx on Pelham Pkwy which is quite reasonable ($800/month full time, with part time options), Bright Stars, but unfortunately they only start at 18 months. I would check the OCFS website (
New York State Office of Children and Family Services (OCFS)) to find listings of licensed family daycare providers and then call around. In my experience, there is one in almost every building. They are safe and nuturing, though basic (little to no outdoor time, a good deal of TV time which I don't love), but your child will be safe and cared for and most take young babies. I've found rates around 900/month can be worked out, particularly if you don't need full time 7-6. ACS also provides childcare vouchers to working lower-income families. You don't need to be on welfare. You can look at this site to check to see whether you qualify:
Child Care & Head Start. Lastly, when I needed care for my infant I ended up using care.com to find a nanny, and I split the cost with another family. For both kids, it was $15/hour. For a full 40 hour week of care, that equaled $300/kid. Not great, but not terrible, and it gives you great flexibility and attention. You may be able to find someone for less - maybe as low as $12-13/hour, split between two.