I am full time student in NY. I did my internship in Seattle and a semester exchange. So, I spent less than 183 days in NY. Do I still need to file NY state taxes?

Normally you report all your income to your Resident State, and get credit for any taxes paid to any other state.

If you are a New York Resident (home you return to when not in school, Driver's License, bank account, etc.), this would apply to you. If you are a 'Resident' of a state, time spent away at school, for a job, in the military, etc. is considered a 'temporary absence'. 

If you were paid on a 1099-Misc with an amount in Box 7, you could report your income as Self-Employment income and claim expenses against it.  Type 'schedule c' in the Search window, then 'Jump to schedule c' to explore this option.

If you received a W-2, and no state tax was withheld, then yes, you would owe tax to your Resident State on that income.

You may want to consider using the  Free File Program since your situation is more complicated, and Free Edition may ask you to Upgrade. 

More info:

https://ttlc.intuit.com/replies/3301816