India's ministry of external affairs said no information was available on the identity of the crew members, but confirmed that 13 were alive.
The ministry said it was in touch with the embassy officials in Yemen.
Saudi Arabia is leading a coalition fighting rebels who took control over the capital, Sanaa, last September.
One of the boats was travelling across the Red Sea from the Somalian port of Berbera to Mokha in Yemen when it came under fire on Tuesday, a foreign ministry spokesman told news agency AFP.
Yemeni coast guard officials said a Saudi-led coalition attacked more than five boats off the Yemeni coast.
Officials with the rebels, known as Houthis, said the boats were carrying fishermen and were not connected to them, the Associated Press reported.
It was not immediately clear if the two incidents were the same.
In recent months Yemen has seen conflicts between different groups, pushing the country "to the edge of a civil war", according to a UN official.
The main fight is between forces loyal to the beleaguered President Abdrabbuh Mansour Hadi and those allied to Zaidi Shia rebels known as Houthis, who forced Mr Hadi to flee Sanaa in February.
Seven Indian nationals are reported to be missing after Saudi airplanes bombed two boats carrying 20 Indian crew members near Yemen.