It's weird, a lot of people who are broadly on the left seem to reflexively support the EU when it's not, when you get down to it, a particularly socialistic project. I think it's just because both small- and big-C conservatives in Britain tend overwhelmingly to be euroskeptics, so if you think of yourself as a leftist you can easily fall into thinking that the only sensible position is a pro-EU one.
My dad loves to bang on about how the UK is a net contributor to the EU but he reads the Torygraph and probably doesn't appreciate that being a member massively facilitates trade with the rest of the EU, which buys about half of all our exports. I don't think successive governments would have allowed the UK to stay in the EU if it wasn't good for our economy on balance.
As Rich says, there's a huge problem in the unaccountability of those who hold power and make decisions in the EU, and I think the left has been foolish to allow the right to make this issue their own personal bugbear.