Yes, but WHICH border?<quoted text>
Ideally, non EU members should be checked for Identification at the border. Many countries let them in so they can spend money in the country. Illegal immigrants don't come empty handed. They come with 3000 or 5000 euros in their pockets. And who knows if they are just transitting and the final destination is some where else.
Can France, Belgium or Germany really trust the border 'checks' operated by countries like Romania, Bulgaria, Greece or even Italy and Spain?
These countries know fully well that the illegals will be heading for Northern Europe after they enter the Shengen zone, and have no incentive to dedicate a large budget to border control.
But, since the majority of illegals are failed assylum seekers who overstay after their cases have been dismissed, why no repeal the assylum status altogether. It's the biggest scam existing, everybody knows it.
Also, illegal entry should be criminalised. At present, we are far too lenient.
I am afraid that to really tackle illegal immigration Western countries will really have to adopt draconian measures, like in good old Soviet times! Until then, politicians pay lip service to it.