Rembges, 37-year-old, began her campaign after finding a family sitting on the street outside Basel's asylum registration centre on a bitterly cold Sunday.
They had just been told there was no space for additional refugees, meaning they would have to spend the night outdoors.
Rembges argued with the security guards, who finally granted the family admission. But the next day, they were back on the street, so she tried to find a better solution.
She sent e-mails, called friends and posted on her Facebook account that there was a family in need. Soon, 25 people had offered to help and nine people were quickly sheltered, Tages Anzeiger reports.
The six members of the African family -- three women, two children and a young man -- ended up in the home of a well-known migrant rights activist.
“Our guest room is quite spacious,” Anni Lanz told newspaper Le Matin.
Since the Eritrean family's situation is no exception, Rembges regularly patrols the asylum centre in Basel to see if there are other refugees in need.
To help coordinate the operation, she has set up an account with Doodle, a free online scheduling tool. Now, when asylum centres are full, she posts a message on Doodle seeking the assistance of friends and relatives.
According to humanitarian organizations, many asylum seekers have been turned away from official shelters in the cantons of Basel City, Vaud and Ticino, despite recent sub-zero temperatures.
Some have been welcomed into private homes or have been put up by the Salvation Army, but many others have had to sleep in cold train stations.
Swiss television show ’10 vor 10’ reported that a Basel asylum centre had been forced to turn as many as 20 people away. The centre already gives shelter to 500 people despite only being kitted out for 320.
"We have no choice," said a spokesman for the Federal Office for Migration, who admits the situation has been particularly difficult in Basel over the last month.
Roger Lang, director of the asylum centre in Basel, also confirmed the problem.
"It's true that we are overwhelmed, but for women and children, we always find a solution," he told La Tribune de Genève.
Switzerland has seen asylum requests soar this year in the wake of popular uprisings in North Africa and the Middle East. By the end of November, the country had received 20,000 applications, 5,000 more than in the whole of 2010.