I'm of the opinion, probably the minority opinion, that the United Nations is not a particularly effective or functional organization. It does serve as a forum for speeches and meetings and it does promote the appearance of world cooperation, but in the most practical ways it is not particularly useful body. (While I'm at it, I'll add that I'm dismayed at the few new initiatives it is promoting, which attempt to create one rule of humanity and thinking for all, something which necessarily limits the role of parents and individuals and giving more control to government. However, I'm sure others think this is a terrific idea.) As to paying dues, it's like anything else--if no one enforces the rules, soon no one will obey them. The U.S. does, indeed, owe dues to this body, and if the rules are going to be enforced for all, we should pay. I'm confident we have paid significantly more than our share (in blood or in treasure) of anything done by the UN since its inception. Enforcement at this point, after years of letting it slide, would be impossible, it seems to me.
The hallmark of the United Nations is international cooperation and participation, so the UN is reluctant to pursue sanctions against any country unless international security is threatened, such as in the case of North Korea or Iran's pursuit of nuclear weapons.
In the case of a nation not paying their dues, we have to remember that many of the member countries are poverty stricken, and choose to pay their dues in UN soldiers' participation or some form other than money. It's hard to go after poor countries for not paying dues when they also can't pay their own bills. In the case of the US, we are also on the Permanent Security Council, which means our country has veto power. If, in some way the UN decided to place sanctions on us, the US Ambassador to the UN could simply veto the resolution.
So, what the UN should do is not necessarily relevant. There's not much they can do.
This is a very hard issue and that is probably why the UN has such a hard time with countries like the US refusing to pay all or part of their dues.
You would think that the best thing to do if a country does not pay its dues is to throw it out of the UN. This is what would happen if someone did not pay their dues or fees in real life, right?
But think about what would happen to the UN if they did this. If they threw the US out, for example, the UN would become pretty irrelevant because the US is still the most powerful country in the world. UN decisions that did not have the US's support would be pretty weak indeed. If the UN started ejecting other countries, it would still harm the organization's ability to do what it is supposed to do. It would presumably lose influence in the world if it started to lose members.
So I don't really see that the UN can do anything except try to persuade the countries to pay up.