I work for a company that was in Houston, TX but your job was in Rochester, NY. They never deducted state taxes nor paid those to the state.
Is my employer responsible for filing NY state taxes?

If your work location is NY then your employer is responsible for withholding NY state income tax from your wages, but the ultimate responsibility for filing and paying taxes is yours (assuming you are a W2 employee and not a contractor). If they didn't withhold from your pay check then you will need to file quarterly estimated tax payments with NY, and of course file a tax return at the end of the year. And you should contact the payroll department ASAP and have them start withholding NY State income tax from your wages.