I was advised to only water my banana trees once a week in the cool season but now that many are flowering and producing fruit my brother- in- law says I should increase the watering.

Is he right? 

Thanking any useful replies in advance.

Your BIL may be right, flower and fruit production requires adequate soil moisture.  But this should be based on a case be case assessment. Get down and dirty and get to know your soil and how it retains moisture, or not. If you don't have a soil moisture meter or a soil sampling tube (both are essential assessment aids that I always carry when doing tree health care inspections), then you can dig down with a shovel in a few spots to see how wet the soil is. If it's muddy wet then don't water. If the soil is drying out in the top 20 - 30 cm, then flood it. In cool weather the soil will dry out slower than in the hot season, so adjust your watering accordingly.

 

Also be sure that you are providing adequate, balanced nutrition. Flower and fruit production takes a lot of Calcium, Potassium and other nutrients.