Welfare works if it is a well-designed system. The problem is that there are many "broken" welfare systems. A welfare system needs to be able to educate the poor, help them get and keep jobs, and prevent recidivism. In order for this to work, there has to be an exit criteria and the program has to be effective enough to give people the skills they need to remain employed and productive.
As some others have said, I would need clarification of what you mean by it working to really provide even my opinionated answer. I think it does work sometimes to provide a social safety net for the disabled and people that truly cannot do for themselves and are in communities that lack the resources or the willingness to take care of them. Of course it is also taken advantage of by people looking for loopholes in the system.
Does it also not work? I would say certainly, it places incentives on things that shouldn't be incentivized like having children that you don't necessarily want or are able to provide for.
I would say that yes, welfare works in an effort to provide temporary assistance to those in need of help due to circumstances they may have had no control over. The biggest problem I see is that for some it becomes more than temporary and becomes a way of life. I do not think that we should fault those individuals for that, I think it is a fault in our welfare system.
The welfare question is so emotionally and politically polarizing that it is difficult to say for a certainty that it "works." I see no reason to debate the meaning of the word; I am sure the poster meant the welfare system in this country. Having said all that; my feeling is that yes, it does work. There are many people who, through no fault of their own, are in need of a helping hand from time to time. At the same time, there are those who abuse the system. There is, however, a tendency to fault the entire system and all its participants because of the abuses of a few. An analogy might be those who refuse to fly because airplanes crash--yet statistically one is seven times safer on a commercial flight than in an automobile. We should not blame the entire system or relegate it as wasteful because it has been taken advantage of by the few.
While I have never been on welfare, there has been a time in my past where I have received assistance such as WIC (Women, Infants, Children) program. We have also received government commodities. As with any "free" assistance programs, you will have some use them for what they are intended, and you will have some who take advantage of it. This is the same situation with unemployment or grant money for college students. Perhaps greater accountability is needed for those who receive assistance.
This is a very interesting question, and your answer to it will depend on where you sit politically. Whilst #2 makes a number of excellent answers, traditionally those on the right argue that welfare does not work, because it encourages poor people to stay out of work and gives them no reason to try and get ahead in life or get a job. Those on the left argue that welfare is essential to help support those that, for whatever reason, struggle financially. There must be a safety net as part of having a civilised society to support those individuals that need it.
It is not possible to really say for sure if welfare works.
First of all, it is necessary to define what exactly you mean by welfare in the first place. Do food stamps count as welfare? What about heating assistance? So defining welfare is important if we are to determine whether it works.
Second, what constitutes "working?" Does welfare work if its recipients achieve an income higher than the poverty line? Does it only work if poverty rates for the country as a whole drop? Does it only work if the people who receive it go on to get jobs and stop needing welfare? There needs to be a good definition of "working" if we are to determine whether welfare works.
That said, I would argue that welfare (defined as transfer payments to people who are below a certain level of income) often does not work (if working is defined as getting the recipients out of poverty). In order to escape poverty, people generally need a lot of things. They need better education. They need access to jobs. They need childcare so they have somewhere to leave their kids when they go to work. Most of these things are needed to get people out of poverty and welfare programs cannot provide all of them.