This data will mostly likely not be on-line. On-line data for New York State is scant and this is relatively recent in genealogical terms.
Where was the couple living in 1930?
In the 1940 Census, where does it say Joseph Crawford was living in 1935?
Was it the same place, Taghkanic, Columbia, New York?
Googling for "vital records" in x county and x state will bring up the site of the county and mostly likely have info of how to obtain a death record.
A death record is a good place to start, however the data on it secondary, and only as good as the informants' knowledge and memory. A primary source of parents' names can be found on a marriage record.
If you know where and when the couple married, this should be the next record you should seek.
I looked through your posting history, it appears you've posted about this line before, and received some good leads. Have you followed up on those?
This is the death index record for the mother of Sarah (Kitson) Crawford, correct?
You should order this dc, this may have her maiden name.
Kitson, Hannah, 26 y, (dod) 1 Sep 1880, (cert#)358210 Manhattan
Click this link: http://www.italiangen.org/NYCDEATH.STM
, you will find a printable form to request the dc, use the cert # above.