If I file for child support can my son's father get any sort of custody?

I have a 3 1/2 year old son whose father has never even met him and I want to file for child support. My fear in doing so is that his dad will try to get some sort of custody in spite of me asking for support. Eventhough he has had ZERO involvement in my son's life will a court grant him any sort of custody just based on a DNA test?? Like I said he has never even met his son?? Any advice or info is greatly appreciated!