The issue in the US is that there is a genuine ideological split between the two parties about what the US government should look like. The Democrats want a big government, which takes in a lot of taxes and provides a lot of services to everybody. The Republicans want a small government, which takes in minimalist taxes and provides minimalist services. (I'm not interested in debating the relative merits of these two ideologies today.) And this isn't just the parties' ideologues, either -- the same ideological split genuinely exists in the US public (contrast to many other countries where the "big government" types seem to have largely won the argument).
Either party could easily balance the budget on its own, given unlimited power. However, because of the way the US political system just happens to work, it's very hard to do anything without getting at least some of the other guys on board.
The Republicans have for many years had a "starve the beast" strategy, figuring that if you keep on cutting taxes then eventually the Democrats will have to agree to cut spending as well. In recent years the Democrats have invented the opposite "gorge the beast" strategy, in which you figure that if you keep on increasing spending then eventually you'll be able to pressure the Republicans into agreeing to tax increases. And the poor beast being starved at one and and gorged at the other starts to... wait, no, this analogy is breaking down.
What's the solution? There really isn't one. The ideological split in the American public is real and it isn't going away any time soon, and folks need to stop proposing "solutions" which basically just come down to "all we need is for the other side to come over and agree with us and it'll all be rosy." That's not gonna happen.
But perhaps the first step is to acknowledge that this ideological split is genuine, and real, and does actually reflect the ideological split within the US population. Perhaps we can compromise, but first we have to acknowledge precisely what it is we're compromising about.
edit: Wow, I can't believe this got modded down. I guess the true believers in the "No really, it's the other guys' fault" die hard.