But the river has a treacherous current and fighting is raging nearby, so many fugitives do not complete the journey. A crowded barge sank outside the northern town of Malakal at the weekend - apparently accidentally - drowning 200 people.
Some 85,000 people have arrived in Minkamen in the past four weeks, overwhelming a resident population of about 60,000.
South Sudan's civil war is exactly a month old and already one person in every 20 is a refugee within or outside the country. Some 400,000 are "internally displaced"; another 75,000 are in refugee camps in neighbouring states.
Most tragically of all, this calamity is being wreaked upon a people who have already suffered beyond measure. South Sudan fought a brutal war for five decades, claiming perhaps two million lives, before finally winning independence from its northern neighbour in July 2011.
This is the world's newest nation and its people surely hoped that their sacrifice might bring some reward. Instead, their country has torn itself apart within three years of its birth.
The confrontation pits Salva Kiir, the President, against a rebel army led by Riek Machar, a former vice-president who was sacked last July. These bitter rivals agree on nothing except that their war is a struggle for power. Machar accuses the President of trying to become a dictator; Kiir says his unruly subordinate started everything by trying to carry out a coup on December 15.
Not many impartial observers believe the coup story. On the contrary, some think that Kiir has been searching for a pretext to deal with Machar, especially since the latter declared his aim to run for president in next year's election.
These two have very old scores to settle. Both fought for independence as senior commanders in the Sudan People's Liberation Army (SPLA).
Machar had aspirations of his own. In 1991, he broke away from the SPLA and tried to topple its then commander, John Garang. The rebels then split along tribal lines, with Machar's Nuers and Garang's Dinkas turning upon one another.
The Khartoum regime, presented with the spectacle of its enemies obligingly tearing themselves apart, duly fanned the flames. Machar took weapons from Khartoum, allowing himself to be used to divide the struggle for independence. He needed 10 years to realise how he had been manipulated, eventually achieving a public "reconciliation" with Garang in 2002.
But Garang died in a helicopter crash a few weeks after the war against Khartoum ended in 2005. Machar might have buried the hatchet with him, but never with Kiir, who took over as SPLA leader and became South Sudan's first president.
Of all the wars in recent history, this one must rank among the most futile. But that makes it no less dangerous. Behind it all lies tribal rivalry between the largely Nuer rebels and the Dinka-led Government.
The bloodshed might have started as a political power struggle, but it is now escalating into an ethnic war.
The refugees who make the journey across the White Nile from Bor are mainly Dinkas, fleeing the predominantly Nuer insurgents who have captured the town.
The only good news is that aid is beginning to arrive. Oxfam is supplying the refugees around Minkamen with 350,000 litres of safe water daily and the Red Cross has begun distributing food.
Nonetheless, Ferran Puig, Oxfam's associate country director, is struck by how "in terms of shelter, there is nothing - people are under the trees".
At night, the sound of artillery echoes across the river, reminding the refugees of the dangers they have fled. Their greatest fear is that South Sudan's war will become an endless cycle of tribal violence. The signs are this juncture may already have arrived.
