The ICRC described the port of Hudaydah as a "lifeline" for the people of Yemen
The International Committee of the Red Cross (ICRC) has had to halt deliveries to the Red Sea port of Hudaydah.
It said this was partly because it had not received security guarantees.
The port has also been targeted by warplanes from a Saudi-led coalition which is backing Yemen's government in its war with the rebel Houthi movement.
UN attempts to bring in five cranes to replace those damaged in the attacks have not been successful.
The ICRC's Middle East regional director Robert Mardini warned that Yemen, which depends on imports for 90% of its food supplies, was now living on its "reserves" and that the "day will come very soon" when they run out.
He said that despite the focus on the damage and casualties caused by the fighting, thousands of people were dying "silently" of complications due to malnutrition, or of minor illnesses or injuries which were not being treated at the 45% of health facilities that are still functioning.
The conflict, which escalated in March 2015, has left at least 7,500 people dead and 40,000 injured, according to the United Nations.
But almost 19 million people are in need of some form of humanitarian assistance.
Seven million people do not know where their next meal is coming from and more than three million, including 2.1 million children, are suffering from acute malnutrition.
The UN's emergency relief co-ordinator, Stephen O'Brien, called on the warring parties to guarantee more access to ports to let in food, fuel and medicine imports on Monday.
Speaking to reporters in the government-controlled southern port city of Aden, he also said the international community needed to "step up its funding".
Earlier this month, the UN appealed for $2.1bn (£1.7bn) to provide life-saving assistance to 12 million people in Yemen over the next year. But it has so far received only $43m.
Almost 19 million people in Yemen are in need of some form of humanitarian assistance
On Tuesday, Mr O'Brien was forced to cancel a visit to the city of Taiz, which is besieged by rebel fighters, reportedly after his convoy came under fire.
In a separate development on Tuesday, UN human rights officials said they had evidence of the recruitment of child soldiers in Yemen, mostly by groups affiliated with the Houthis.
In all, the UN managed to verify the recruitment of 1,476 children, all boys, between March 2015 and the end of January 2017. However, it said the numbers were likely to be much higher.
(c) 2017 BBC News Middle East