The boat was carrying 53 people up the Kapuas river Sunday when it hit a log and sank, Slamet Riyadi, the head of the local search and rescue office told AFP.
As well as the baby, three women were killed.
"We have been searching the river for the 10 missing people since yesterday but so far we have not found anything," Riyadi said.
Among the missing were three children, aged five and six.
Indonesia relies heavily on boats to connect its more than 17,000 islands, but has a poor maritime safety record.
Borneo, the world's third-largest island, is shared between Indonesia, Malaysia and Brunei.