I am a New York resident. For 2019, I went to basic training out of state on 01/05 but did not complete the course and returned to New York on 03/01. I went back to the same course out of state in 09/07.
Do I qualify for part-year residency for my NY income earned out-of-state in the military which I was in for about 6 months?
You are a full-year New York Resident. With military resident rule, your SLR (State of Legal Residence) does not change when you are stationed outside of the state. All of your income is subject to New York tax.
However, New York does provide an exclusion of active-duty military income from state tax whenever you are:
To maintain a residence, you must live in an independent living situation (a home with a mortgage or an off-base apartment you sustain). Living in the barracks is not considered independent living per New York law. For more information, please see this webpage: Information for military personnel & veterans