My employer is in NY but I live in NJ and work in NJ on-site at one of their clients. For the past 3 years they have been deducting NY taxes which has cost me money at tax time. When I asked them about this they said they didn't think they could retroactively fix it (they finally corrected it for this year). Are they responsible to correct it since they said I was working in NY when in fact I was working in NJ?

You can correct it yourself - but it may not be worth it to you. For the past 3 years you should have been filing NJ taxes on your wages, and also filing a non-resident NY tax return that shows your wages allocated 100% to NJ. You do that with New York's form IT-203 and IT-203-B. That way you would have gotten all your NY tax withholding back as a refund, which you could have applied to your NJ income tax. But be aware that if you try to fix this now by filing corrected income tax returns for both NJ and NY for the past 3 years you will get a big refund from NY but you will also owe NJ big income tax payments es for all three years plus interest for being so late on filing and payments. I would advise that you let sleeping dogs lie.

Dude! You need to walk up to those tax people and complain because if you complain and don't give up you eventually get that money back.