Say it's May 1999. I usually just put 05/01/1999 or 05/15/1999 if I know it was in the later part of the month. I do my best to get as close as possible. But this is as a US citizen and applying for jobs. I'm not sure if it's acceptable to do otherwise?
I graduated high school on May 19, and remember it specifically, yet my transcripts say June 20th, as this is when my diploma was awarded. I have never had this be an issue though, as long as you get the month and year with the date as close as possible, I believe you would be fine.
To be sure, contact the BON in NY and see what they have to say. Good luck.