Brown Out Problem in Raspberry Pi Over Power Tether Cable
The problem you are experiencing is due to the resistance of your umbilical.  You have a transmission line.  The \$I^2 R_{Cable}\$ losses are eating up your power.  

Measure the voltage and current at the terminals of your motor.  As I increases \$V_{Load}\$ will decrease.

There might be enough voltage to start at no-load (especially if rov is out of water - pushing air) but as current increases, voltage drop to wire increases, which explains your observed characteristics. 

Solution would be to send ac at a higher voltage to rov, step it down in the rov and drive motors.

Edit...
Or increase the gage of your power feeder cable as suggested by: Tether - Power and Voltage Drop Examples.

It really depends on your current.  6 motors can draw a lot especially going from locked rotor to full speed.  20A is a lot depending on length of your umbilical.  

Look up resistance of wire from AWG (American Wire Gage).  They have R/100ft.  Multiply it by your tether length * 2 / 100ft.  

The 2 is because you have to get current to load and back.  Think about number of pins on a lamp.

Or you can also calculate resistance.  $$R_{Cable} = \frac{ρ l}{A}$$
ρ = 10.37 Ω CM/ft, l = 2 * length of your umbilical, A in CM from AWG.

$$V_{DROP} = I R_{Cable}$$

Components of an ROV system