Siôn Simon, MEP for West Midlands which covers Shropshire, said part of the blame for that lies with those campaigning to stay in who have tried to avoid confrontation over it.
Mr Simon believes Britain will be hit hard financially if it leaves the EU following the referendum on June 23.
"I don't think we are getting as good publicity. I think there are a lot of reasons for that," he said.
"I think it's partly our own fault. There's been a lot of hostility in this country and I think that those who have been in support of it have tended to just keep our mouths shut and try and avoid having a big row about it.
"It's a bit late for people to say it's a positive case. If you have not been positive for 30 years but you have let people be consistently relentlessly negative for 30 or 40 years it's a bit much to expect people in the last few weeks before the referendum all of a sudden to be marvelling at how wonderful the European Union is.
"I think we have won all the arguments. What worries me, if we are not careful, as a nation we are sleepwalking into something that will damage the UK, will damage our economy, and damage the daily lives of ordinary people in this country in ways that people are not expecting.
"People think we can just leave the EU and it will not make a difference. People say they need us as much as we need them. They don't need us as much as we need them. We need them a lot more than they need us." Mr Simon has also accused key figures wishing to leave the EU of lying to the public.
"I don't think the campaign has been very edifying. I think there have been terrible lies by the leave side.
"Maybe the remainders have exaggerated, but the leavers have told bareface lies in a very deliberate and systematic way.
"If we leave the EU there will be an immediate increased recession in this country. People will lose their jobs.
"We have had a lot of economic benefits for a long time. Why take a huge risk for something we don't know?," he added.