ATHENS—The risk of deaths among refugees and other migrants crossing into Greece by sea from Turkey, a key passageway in their bid to reach Europe, is expected to rise sharply in coming months with the onset of winter, volunteer groups and Greek officials warn.

Greek officials are bracing for more migrants, most of them refugees escaping conflict and violence in Syria and Afghanistan, to cross over into one of the country’s islands, undeterred by the worsening conditions.

Many of the modest measures announced last week by the European Union to help fight the refugee crisis may not be implemented in time to prevent a further rise in the death toll, leaving Athens to fight a growing problem with limited resources. The EU measures include more money for Syria’s neighbors to help settle refugees from the war-ravaged country as well as more funding for United Nations relief operations.

With the arrival of fall, the weather over the Aegean Sea separating Greece from Turkey is already worsening.

Gales force winds, along with a drop in temperature, are hitting the area that sometimes sees snowfall in winter. But the number of those traveling across in inflatable dinghies remains as high as in summer.

In the latest deaths in the Aegean, at least 17 refugees drowned Sunday when their boat sank near Bodrum in southwest Turkey, on a failed attempt to reach the Greek island of Kos, according to Turkey’s state news agency Anadolu.

On the island of Lesbos, where half the refugees entering Greece land, officials report several thousand arriving every day in a trend likely to continue.

“If people continue to make the trip as we anticipate they will, we could very likely see more deaths,” said David Miliband, head of humanitarian relief group International Rescue Committee. “We think this is indicative of the degree of despair setting in in Syria.”

The death toll in the eastern Mediterranean has already surged. More than 70 people died at sea in September—the deadliest month for refugees trying to cross to Greece in two years, according to the International Organization for Migration.

“Even in amazingly strong winds, the traffickers put people in the boats when it is almost certain that they will sink,” said Laura Pappa, head of Greek aid organization Metadrasi.

For many of the migrants, the dangers of winter won’t end with the sea crossing. With risks of hypothermia rising, some may not survive the trip to their final destination in Northern Europe, which can take up to several weeks.

“I am certain that a lot of the children die along the way,” Ms. Pappa said.

Seeking to avoid the dangers at sea, more migrants are heading to Turkey’s western frontier province of Edirne, where they hope to continue overland to Greece or Bulgaria but have so far been prevented from crossing the border by Turkish authorities.

About 350,000 refugees and other migrants have entered Greece this year—of which 70% arrived since July. The problem is hard to manage for Greece partly because it changes so rapidly.

“Whatever contingency plans had been drawn up were changed and changed suddenly,” said Rear Adm. Ioannis Karageorgopoulos, the coast guard’s director general for security and law enforcement.

Tracking migrants entering Greece is proving trickier than on the Libya-Italy route, where traffickers use fewer and larger vessels to transport people.

In the eastern Mediterranean, where Greece is separated from Turkey by as little as 5 miles, local authorities are faced with as many as 10 boats traveling to any one of 22 Greek islands all at once—each carrying up to 50 people.

Greece is facing criticizing from Germany and other European countries for not doing enough to control its borders. Greek authorities say they need practical help from the rest of Europe to help alleviate the cash-strapped country—struggling for years with a severe debt crisis—of the burden of being Europe’s main gateway for a movement of people on a scale not seen anywhere globally since World War II.

“It’s not just about the money. It is about assistance in kind. Even if we are given more money now it will take at least two and a half years to take delivery of any new boat,” said Rear Adm. Karageorgopoulos. “It’s now that we need more planes and boats, and not later.”

The quickest way to make Greece’s borders safer is by increasing patrols with the help of equipment and crews sent by European peers, Greek officials say.

Countries such as Norway, Latvia and Italy have contributed so far to this effort while other larger EU peers such as Germany and France have yet to do so.

Greece’s economic woes have grounded most of its coast guard’s helicopters and planes and almost a third of its 240 vessels.

The government Athens is throwing more resources at the problem, under the guidance of the country’s lenders—other eurozone nations and the International Monetary Fund.

In 2015, the coast guard’s operational budget was increased by 50% from last year to €15 million. The government has also relaxed the rules on a hiring freeze, bumping up the coast guard’s personnel by 200 members to more than 7,000.

But the extra officers won’t be ready to hit the seas for at least another year because of training needs.