Money has always been a disputable topic. Viewpoints differ depending on the financial status of the persons involved in the discussion - however, deciding whether it is impotant to be responsible with one's money in general is essential.
I, personally, think it is important to be responsible with your money, as, in my opinion, money is monetarized energy. Energy of people, who earned it; energy of goods and services purchased for this money. And, according to the law of conservation of energy, the person dealing with money needs to keep the energy flow constant. Being responsible with money means being responsible for the overall financial balance of the economy.
Firstly, the person must be aware of basic economic terms, such as inflation and consumer basket. Handling money responsibly means allowing the economy work smoothly, in correlation with its natural laws. Secondly, it helps curb the rocketing consumerism, as people start spending smart. And, finally, it may bring about changes in human values - knowing that money is just a qualitative measure of energy allows to value real people and events more.
There might be, however, other points of view. For example, someone may think that money is a volatile item and shouldn't be taken seriously. And there are 2 possible evidences to that: nowadays there are many on-line and off-line programms and tools to plan and track the financial situation - which helps the person to stay out of boring calculation and monitoring process; and, in addition, not all people are economically literate and patient enough to be invlolved into budgeting (even if it's on the household level).
But those are quite arguable. It's important to understand that, when it comes to key matters (such as money or health), it's very risky to rely solely on products of IT industry. Besides, in modern world getting basic knowledge in economy and fincance is easy and affordable for everyone.
So, in conclusion I'd like to paraphrase Benjamin Franklin and say: "Teach your children to save money. They'll learn the art of spending themselves". Being responsible with money is a significant step towards financial and social freedom.