The Limits of Government - helping the poor?

Libertarians and liberals (in the classical meaning of the word) are people who want limited government and unlimited individual freedom (as long as no one else's freedom is being limited). The general idea, based on some experience, extensive logic and many facts, is that a Minimal State is the only fair and just form of state, and that the State's main function is to stop violence, make sure contracts are enforced and protect lives of individuals along with their property...

...and make sure that the poorest in the society are helped with some kind of a security net funded with taxes and operated by the State (either directly administrated by it and funded with taxes, or just the latter).

Or what?
The last statement bothers me. Is it really the State's responsibility to make sure the poorest of the poor have somekind of safety-net? I admit that the thought sounds right, and this is what I've heard from even the most liberal of libertarians (excluding anarcho-capitalists of course). But still the statement bothers me.

The Problem
The State is in an awkward situation when it comes to defining those who need help, and those who don't. Also, the State cannot allow access to tax-collected funds to just any-one who makes a strong enough case and does a good job convincing the social servant of some needs for help. The State must treat everyone equal and that means making general rules which again means putting different individuals with different problems into similar boxes which get similar help.

This is not what the hungry, the sick or the unlucky needs to improve his or her situation. Those who need help need it on individual basis. They need help - not welfare-checks or forced institutionalization. Only private charities and organizations can meet the individual needs of the individual because they don't have to follow a general set of rules which have to treat everyone the same.

The States responsibility
That being said I must say I oppose State-supported welfare-programs, even those only intended for the most poor or the most sick. However, this is not to say I oppose that it is the societies responsibility to make sure no-one dies on the streets from diseases, hunger or whatever it is. I think it's the societies responsibility to take care of those who need help, and show understanding for the individual's individual circumstances!