So let's take a look at your sentence:He received a medal for sinking an enemy ship and the capturing its crew
This sentence is not grammatically correct. Notice that the two underlined sections begin with different words. The insertion of 'the' at the beginning destroys the parallelism. By simply getting rid of 'the', you will get two parallel simple gerund phrases:
sinking an enemy ship = capturing of its crew.
That does not mean that you cannot start a phrase with 'the.' However, if you start one phrase with 'the' then you have to start the other one with 'the' as in:He received a medal for the sinking of an enemy ship and the capturing of its crew.
Hope that helps
Magoosh Test Prep