Mostly because it is a system that the legislators that we voted for created and voted for. Employers pay an extra tax to both the states and the federal government to fund unemployment. During good times the unemployment fund runs at a surplus, and thus creates some intergovernment debt which I believe may be included in the national debt, during bad times the unemployment fund may run at a deficit and draw from the past surpluses.
We could end the unemployment benefit, but I think it would only be fair to stop charging employers the extra tax that they have to pay into the system, thus in the long term, there might not be any net reduction in the federal debt due to eliminating unemployment.
Unemployment benefits are also sort of an automatic stimulous program in that when unemployment is low, the amount paid out is fairly low, and when unemployment is high and our economy needs some stimulous, the total amount of unemployment benefits paid out is higher, thus stimulating the economy more.
If nothing else, unemployment benefits reduce other benefit expenditures that the gov would have if we didn't have unemployment, such as foodstamps and welfare.