Getting married..Can I be held responsible for his unpaid back child support?

What is the name of your state? Montana

My fiance has a back child support order from over 10 years ago that we just became aware of. His children are in their late 20's now. If I go ahead with the wedding, can they come after my earnings and property?

You have to be careful here. You must keep everything separate, including checking accounts, savings accounts and the title of the house. Anything with his name on it can be siezed, garnished, or a lein placed on it.

Is he paying at all or is he just blowing this off?

In florida you can file for injured spouse. Anything that is yours can not be touched especially income taxes.