The IS group has seized large parts of Iraq and Syria, declaring a "caliphate", imposing its harsh interpretation of Islamic law and committing widespread atrocities including beheadings and crucifixions.
The Syrian opposition has warned of potential massacres should IS extremists advance on Ain al-Arab, and pleaded for international intervention.
But despite US promises to expand its air campaign against IS in Iraq to Syria, there were no signs yet of American strikes in the country.
Speaking of his decision to flee, Sahab Basravi said: "When the Daesh (IS) attacked Ain al-Arab, we were frightened. They said in the mosques that they could kill all Kurds between seven and 77 years old. So we collected our things and left, immediately."
He said he had not witnessed fighting or executions, but the reputation of IS jihadists was enough to persuade him not to stay.
UNHCR said it feared the massive influx of refugees would only grow, and said authorities were preparing for the possibility of hundreds of thousands of additional arrivals.
- Tear gas and water cannon -
IS fighters have been advancing on Ain al-Arab since late Tuesday, hoping to cement their control over a large part of Syria's border with Turkey.
On Sunday, they were within some 10 kilometres (six miles) of the town, after capturing more than 60 villages in the area, the Syrian Observatory for Human Rights said.
The fighting has killed at least 27 Kurdish militants and 39 IS jihadists, according to the Britain-based monitoring group.
At the border on Sunday, Turkish security forces used tear gas and water cannon to disperse a solidarity demonstration by Turkish Kurds and later closed most border crossing points in the area, including one used by Kurdish fighters heading to Syria.
Only two posts remain open, and the interior ministry will now register new arrivals.
Mustefa Ebdi, a Kobane resident, local journalist and activist, said the streets of his town -- once home to about 50,000 people -- were virtually empty.
"Most of the women and children have left Kobane, but there are thousands of Kurdish men who have taken up arms and are ready to defend the city," he told AFP by telephone, adding that the fighters would have a hard time matching the heavy weapons of IS.
"We need one US airplane to strike those barbarians, where is this international coalition?" he asked. "We are waiting for a miracle."
The Syrian opposition National Coalition has urged foreign air strikes to "stop mass atrocities".
US ambassador to the UN Samantha Power told CBS's "Face the Nation" that "no decisions" on US strikes in Syria had been made yet.
Washington has assembled a coalition of more than 40 countries to fight IS, and has already carried out 186 air strikes against the jihadists in Iraq.
Power told ABC's "This Week" that she predicted Washington "will not do the air strikes alone if the president decides to do the air strikes".
As part of the bid to broaden the coalition, US Secretary of State John Kerry on Sunday raised the threat of IS militants directly with his Iranian counterpart in rare high-level talks in New York.
Kerry has said Iran has a role to play, but the country's supreme leader Ayatollah Ali Khamenei had however said last week that his government had rejected Washington's request to join the coalition.
In Berlin, Germany's foreign minister Frank-Walter Steinmeier stressed that his country would not participate in air strikes or ground offensive, signalling that members of the coalition would take on different tasks in the battle against the jihadists.
- Outrage over IS atrocities -
Iraq's defence ministry on Sunday said it had lost contact with soldiers during an operation to rescue an army battalion under attack by militants near the city of Fallujah, west of Baghdad.
Iraqi military spokesman Qassem Atta told AFP that the operation was backed by US air support.
The Pentagon did not immediately confirm the information.
In a statement, the US Central Command said US military forces on Sunday carried out two air strikes near the northern city of Sinjar, destroying an IS mortar position and an IS "semi-tractor trailer carrying munitions".
International outrage has grown over IS atrocities including the on-camera beheadings of two US journalists and a British aid worker.
The wife of a British taxi driver being held hostage by the jihadists urged them Saturday to "see it in their hearts to release my husband".
Alan Henning, a 47-year-old father of two, volunteered to drive a humanitarian aid convoy to Syria for a Muslim charity but was captured 10 months ago by IS.
Turkish President Recep Tayyip Erdogan said Sunday that dozens of Turkish hostages held by IS militants in Iraq had been freed as a result of negotiations and no ransom had been paid.