David Henderson at Econolog talks about the myth of market failure in regards to unemployment insurance. He makes some excellent points and there are more in the comments.
Something not addressed is that unemployment insurance has a superior substitute good.
To understand what a substitute good is, you might consider that attendance at plays goes up when movie prices go up, or sales of margarine go up when butter is less expensive, or cheaper cell phones lead to a drop in people purchasing plain old telephone service.
A substitute good for unemployment insurance is "safe" investments. It would be difficult to pay someone else enough to support you AND monitor that you lost your job and can't get another one through no desire of your own (the inherent moral hazard) and have them be able to pay you more in benefits than if you took your unemployment insurance premiums and simply invested them safely.
Essentially, you are self-insuring. You are the most efficient at that, because no moral hazard exists when you have perfect information on your own motivations and desires.
The next best option is a mutual assistance society that is run by people who know the members and can decided on benefits. That means there isn't a faceless bureaucrat deciding if someone deserves benefits, but rather someone in a much better position to determine exactly what is really needed and exactly how much work the someone is able to do in return for their benefits.
There are still some mutual assistance societies, but many of them vanished when the government "took over" their market and drove them out by using mandatory taxes and deficit spending to under price them. The ones that remain tend to have another primary function.
For example, the LDS church (among many other things) functions as a private welfare society. In their example, they reduce the inherent moral hazard in the situation by having those seeking welfare interviewed by the volunteer leader of their local congregation to determine exactly what help they need, but more importantly what they REALLY need (not money to eat out, or to live in a mansion), and what they themselves can contribute. It's not unusual for an LDS member on welfare to clean up the church grounds or work at a welfare farm if they are able to while seeking new employment, while someone sick may not be asked to do anything at all.
That sort of differentiation and local personalization is what a federal government program is unable to match.
