I live in Washington state and I am about to get married. He has four kids from previous marriage. He does not work in the winter so I am the bread winner during that time. Am I obligated to pay his child support?

Legally, no you aren't obligated to pay his support. However, if you don't want him to go into arrears and he has no personal money to pay the support then you will need to. Was he working the same job he is now when support was calculated?

While they will not come and drag you to court and then jail if you refuse to pay, they might do that to him. His child support does not just go away because he cannot pay now... and it will be there forever. Regardless of the fact that it cannot be legally collected from you, they can go after joint assets (bank accounts etc).