The only way to increase rail capacity to Gatwick Airport is to make all Southern and Gatwick Express trains on the Brighton Main Line 12 cars like all the Thameslink trains. This would almost certainly require an increase in the amount of rolling stock available for GTR Southern and Gatwick Express services and for GTR to acquire more rolling stock in total. It is probably not possible to greatly increase the total number of train services on the Brighton Main Line but the issue can be solved by operating the existing train services with longer trains with a lot more seats. The focus needs to be on the rolling stock used for these train services.
As has been said elsewhere, when they did run as 12 car before (and even before the 387s came in when they were 10 car?) much of the train was empty anyway because people didn't spread out at Gatwick to board the train, and presumably didn't walk down far enough at Victoria.
Now, I am sure Victoria to Gatwick wasn't quite as bad as people may keep walking when the train is busy, but unless the train is on the platform waiting for some time at Gatwick (so people can do the same thing) then it's just going to be busy nearest to the lifts and escalators.
Gatwick also needs to look at the new services it will offer. Again, as has been mentioned before, there will be flights in clusters depending on the origin of the inbound flight and vice versa. There will effectively be peak times that may not match traditional railway peak times, and that's where the capacity uplift will need to come in. With lots of very early flights for short-haul low cost routes, the peak effectively starts much earlier and there must also be quite a lot of busy trains when those same low cost airlines bring people back after 9pm.