A little more information would be helpful. What are you worried about exactly?
But the short answer is that you give your address to your employer when you fill out a W4, and they tell the state. You also tell the state your address when you file taxes. Your landlord may follow the law (shocking!) and put your security deposit into an interest-bearing savings account, which probably would need to be under your social security number. I've heard of landlords giving tenants a W4 form for that purpose...