I was working as a full time employee for a consultanting company in Chicago, IL. While I worked on a project for the client in NY, I flu there every week (Mon-Thur). Do I need to pay tax to NY? The tax software shows that I own them more than $3400 tax, and I have only worked there for about 1 month, is that normal?

The income earned during 1 month period you were in NY is to be reported on the NY tax return. You are part year resident of NY. File NY tax return only if the income (during the days you were in NY-- to can use simple arithmetics to figure out your income).
Also report same income on IL tax return. Claim credit for taxes paid to IL. Your U.S. Tax Return: Working or Living in Two or More states