If Ohio, say, decides that the welfare program is unnecessary, and the program is given at a state by state level, then Ohio can remove welfare completely.
Conversely, if Alaska doesn't have enough unallocated funds to support the welfare system for its citizens, then they will also lose benefits.
Currently, the funds are federal, but the management is by state and county. Each state has guidelines for ensuring their citizens get through and then off cash benefits as quickly as possible. Federal funds preclude any state from denying benefits based on that state's political climate, and state by state management ensures that the funds are looked after accordingly.
As it stands, recipients of any form of welfare, including cash, food, medicaid, and housing, have to prove their income at least once a year for each individual program. That means if an individual receives food, housing, and medicaid, they must certify themselves eligible three times a year. This ensures that those who might try to abuse the system are watched closely. It doesn't stop abuse, but it lessens it considerably simply by making it harder for people to claim benefits in the first place. It used to be that people claimed welfare because it was easy. Now, not so much.
So then we are left with food stamps and housing. Which in my opinion, we cannot do without. The average market rate rent in my area is $1000 for a 2 bedroom apartment. The average fast food worker makes less than $1000 a month before taxes. How do you expect them to live off of that? Especially those who haven't the experience or the schooling to find a job elsewhere.