No, in fact, the illegal alien becomes legal. Anyone who marries an American citizen OR green card holder will become legal. No questions asked.
If you receive government benefits, marrying an illegal alien will not stop that. The illegal alien himself cannot receive those benefits but his US citizen spouse and/or US citizen children can as long as they qualify. This could hurt the process of getting him his greencard in the future, however, since the US citizen spouse has to prove that she can support the alien. If she has been on welfare, she obviously hasn't been supporting him and this could make it so he can't get his greencard. Do not use government benefits unless they are absolutely necessary to feed your children. Try to maintain your family on your own.
When I married my husband they told me I had to sign papers that my family would not be eligible for food stamps. I I think it is an attempt to keep families from working the system which I completely agree with and I was actually quite happy that the government was doing something about the people coming here just to live off the money that we work so hard for.