Ban's warning came as the first U.S. plane arrived in the country, carrying relief supplies for the survivors, who have received little help since Cyclone Nargis ravaged the country on May 2.
Myanmar's official death toll figure is about 32,000 although some estimates say it could top 100,000.
"I want to register my deep concern -- and immense frustration -- at the unacceptably slow response to this grave humanitarian crisis," Ban told reporters in New York, the U.N. News Center said.
There were reports the junta had taken control of items meant for storm victims and were distributing to people unaffected by the cyclone.
"Unless more aid gets into the country -- very quickly -- we face an outbreak of infectious diseases that could dwarf today's crisis," Ban said
The report said about 1.5 million people are at severe risk in the wake of the devastation.
CNN reported a U.S. C-130 Hercules plane carrying supplies arrived at a time when survivors have been forced to pump water from ponds filled with bodies. Two more U.S. Air Force planes were expected Tuesday.

