If you're lucky enough to be in a job where you can join a union you'd be a fool not to quite frankly.
I think that's a large part of the problem when it comes down to public support.
To paint with very broad brushstrokes, in the beginning there was the industrial revolution and mass urbanisation of the UK. That created a huge class of urban labourers who were pretty much universally exploited and subjected to working conditions that had a massively detrimental effect on the quality of life of them and their families.
Then unionism began to grow, especially in the first quarter of the twentieth century. If you worked in manufacture or associated industries, it would probably have been possible for you to be unionised. The amount of good this would have done is obviously up for debate; governments and employers have taken hard stands against unions for as long as they have existed. The point is, though, that unions were formed to address the needs of workers in the industries that employed most people.
Since the mid-twentieth century though, there has been a widespread de-industrialisation in this country. This has led to conflict between unions and government and between unions and companies. But, more importantly, the shift to an economy where a huge proportion of the jobs are in service industries has absolutely not been accompanied by a rise in service-sector unions. So you have a situation where the people in the demographic of lowest-skilled workers (likely to be the people who enjoy the least economic security and lowest quality of life, and thus whose interests are most in need of defending) have gone from being mostly unionised to being mostly without any kind of labour organisation to represent their interests.
Obviously there are all kinds of other developments that feed into this topic (creation of the minimum wage, British and European legislation on working conditions, etc.), but the fact remains that being part of a unionised workforce is a foreign experience for most of the millions of people who are employed in the private sector in modern Britain, and so it's not all that surprising that public sympathy for unions isn't all that forthcoming sometimes. Doubly so in a period when private sector workers have to bear the full brunt of a recession, but where public sector workers (by virtue both of being government employees and by being more likely to be unionised) seem to have an extra layer of protection shielding them from the impact of the nation/world's economic problems.
I think that, for anybody interested in workers' rights and interests in twenty-first century Britain, the key question is how to secure a better settlement for workers who have grown up in a world where traditional trade unions are of ever decreasing relevance to the fields of employment of the majority.