It is early in the afternoon at this important port 125 miles from Athens and two Greek navy officers are patrolling the docks, each wielding sticks with mirrors to peek under trucks. They have seen nothing so far.

Suddenly, three young men burst from behind a massive container and take off down a dock. The officers begin a chase, but the drama is over before it starts. Within a minute, the three men, faster and more desperate, have escaped into a dilapidated industrial complex.

The men, says one officer, are illegal immigrants, who apparently spent the night on the dock hoping to sneak onto a ferry to Italy. They are part of a deluge of undocumented workers trying to reach Europe through Greece, and slipping past authorities is just part of the process. "Day in, day out, the same story," the officer lamented, trying to catch his breath.

The country quietly has become a steppingstone for a wave of Middle East and South Asia workers fleeing job markets ravaged by years of government turmoil. In 2011, an extraordinary year because of the uprisings in North Africa, 140,980 people were detected entering the EU illegally, up 35% from the year before, according to Frontex, the EU's border-control agency. Of those, 40% came through Greece. Through July this year, 23,000 people were apprehended crossing the border illegally, roughly 30% ahead of last's year pace.

Border control in Greece isn't a new problem. But the country's economic malaise and budget restrictions are hampering many of its efforts to reduce the flow of illegal immigration. Hoping to come to the rescue, the Europe Commission—the EU's executive branch—began pouring €255 million ($331 million) into border protection for Greece over the past two years. But that is still less than it gives some countries with far smaller border problems. And whatever it gives, years of bloated bureaucracy and now new public hiring restrictions in Greece have stalled some of the best-laid plans. According to one confidential EU report, the country has hired only 11 staffers to help process asylum cases, despite funding last year for 700 positions.

Add deplorable detention conditions at immigration centers, according to EU officials and human-rights groups, and mounting domestic unrest over the influx of foreign arrivals, and Greece finds itself with yet another Olympic-size crisis.

“Greece isn’t Europe”

In response, government officials say they are doing the best they can under difficult circumstances. The country's new minister of public order, Nikos Dendias, says Greece takes border protection seriously, but that the influx from abroad is reaching crisis proportions. He calls Greece a "buffer zone of Europe" that carries "a disproportionate burden."

Austrian Home Affairs Minister Johanna Mikl-Leitner has described the Greek-Turkish border "as open as a barn door," and her government maintains that shutting Greece out of Schengen should remain an option. Germany, Finland and the Netherlands—three governments that have taken the toughest stance toward Athens in the euro crisis—also have repeatedly voiced concerns about the Greek border. Diplomats say they are likely to back plans to reintroduce passport checks for travelers arriving from Greece, effectively isolating it from the rest of the Schengen zone.

Much of the problem is taking place in Greece's northeastern border with Turkey, an 80-mile fault line dominated by the river Evros that has become Europe's most porous and politically potent frontier. But ports like Patras have become a favored gateway from Greece to the rest of the European Union for many illegal immigrants. From here, they travel across the Ionian and Adriatic seas to Italy, where they will stay or sneak into more countries. In the Schengen zone, travelers don't have to show documentation but some countries are increasingly conducting spot checks to apprehend illegal immigrants when they cross borders.

Here in Patras, the third biggest port in the country, a couple hundred yards from docked ferries, the stench of dried urine under scorching sun announces squatters. By the old train tracks, a dirt path leads to a derelict industrial site. The Piraiki Patraiki factory, once a nationally renowned textile manufacturer, was for months inhabited by dozens of illegal immigrants, according to border officials. Along with Afghanistan and Pakistan, they were from Sudan, Morocco and Somalia, with a few from the Ivory Coast. Though recently cleared by police, some 80 men slept and ate in gutted rooms where cotton was once woven for export.

"Greece isn't Europe. It's Asia, it's Africa, it's not Europe," said 23-year-old Mohammed Ashar, at the derelict factory. A native of Morocco, he paid a Kurdish smuggler in Turkey €1,500 ($1,950) to cross the River Evros, but was apprehended on the Greek side of the border and taken to Fylakio, a detention center nearby. There, he was fingerprinted and registered in the EU border database. On the paper, his date of birth: New Year's Day 1989. Like most people entering Greece, he doesn't know his birthday, so many are registered with a Jan. 1 birth date.

Just one central processing office

The paper also required Mr. Ashar to make his way home after a month—highlighting another immigration challenge for Greece and its EU partners. Across Europe, illegal immigrants may be held in detention centers anywhere from a few hours to a few weeks, then given papers requiring them to leave within a month. That saves on the cost of returning illegal immigrants to home countries, but limits most deportations to those caught staying beyond 30 days.

Like so many before him, Mr. Ashar didn't return home. Instead, he traveled to Alexandroupolis and Athens before coming to Patras, escaping arrest during the police evacuation of the abandoned factory. In late July, after nine months in Greece, Mr. Ashar sneaked on a ferry to Italy, where he hopes to stay. "There are no jobs for Greeks, how can there be jobs for us?" he asks in French-accented English.

Though Greece bears the brunt of the political pressure, it isn't clear the country has the biggest illegal immigration problem in Europe. Last year, some 351,000 people were in the EU illegally by overstaying their visas, more than twice the number of those who crossed into Europe without documents, according to Frontex.

Protecting borders is an issue of national sovereignty most European governments aren't willing to surrender to Brussels. The European Commission is trying act as an umpire to preserve the passport-free zone, but Cecilia Malmström, the EU Commissioner for Home Affairs, said discussions on immigration have grown difficult under the current political mood in member states.

In June, interior ministers from EU governments insisted that national authorities should have new powers to reintroduce passport controls for up to two years to isolate a Schengen country if it has "serious deficiencies" in border control. Their vote has led to a bitter and still unresolved battle in Brussels, with the Commission and the European Parliament—the only popularly elected EU institution—fighting the decision.

Indeed, how the EU has decided to distribute its war chest for border protection has become something of a political football—and a statement about its assessment of Greek use of public funds. While conceding they are handicapped by an inflexible, long-term budget, EU officials say that Spain and Italy combined receive more funding because they are better at handling asylum requests. The entire country of Greece has one central processing office, open for just a few hours once a week in Athens. "In terms of asylum, Greece has no system to speak of," said one EU official, who declined to be identified.

Poor EU cooperation with Turkey

In an internal report last April, the European Commission also faulted the government's legendary bureaucracy for being "too heavy" and "impeding the rapid disbursement" of the EU funding. Greek government officials say they are trying to reform that, but say they also have been hampered by hiring ban imposed by international creditors trying to curb the country's public spending. They said they are trying to establish a more integrated system for processing asylum cases.

At a cafe at the Alexandroupolis train station, a man who identified himself as a smuggler from Algeria, described the illegal-immigration class system. Syrians are the richest and are willing to pay €10,000 to reach Norway, Finland or Sweden for asylum. He said he charges €5,000 for a fake Greek passport and €1,500 to just cross the river.

Bangladeshis are lied to by smugglers about their prospects in Greece, he said. "They tell them they'll be kings in Athens," he said. "But when they get there they see there's no kingdom." The man, who calls himself Nassim, said he has been in Greece for five years, smuggled there himself.

Ms. Malmström, the EU Commissioner for Home Affairs, said trafficking like this into Europe has become a €25 billion-a-year-industry, so lucrative it is hard to combat. Among other steps, she said, Europe is sharing more intelligence among member states, while Frontex, the EU border agency, has become a constant presence at the Greek-Turkish, sharing officers from EU states for monthly surveying missions.

But the issues keep piling on. Among them: poor EU cooperation with Turkey on migration. Greek authorities can't return illegal immigrants to Turkey because the EU and Turkey have no readmission agreement. In June, EU ministers agreed to start talks over an accord. But with Ankara demanding that Turks be eligible for travel visa-free into the Schengen area, no date has been set for a final resolution.

And even within the EU, solidarity isn't the norm.