Maybe?
I think of it like this. When you are starting or running the car and its accessories, you're drawing a certain load through the switch. If the switch, contacts, wiring etc. are too small to easily pass the current needed to support the load, then you have resistance. If the total current supplied to the switch (which a brown wire mod increases) is too small, then the resistance is greater. Resistance creates heat, which damages the switch and other components further... increasing resistance and so forth until failure begins.
So, in short (heh) I think a brown wire mod would help. There seem to be many versions of this mid, the original "brown wire mod" doubled up the wire from the battery to the ignition switch, to help supply power to the switched accessories. This is the version that would help you. Part of the "mod" is just maintenance; inspecting, cleaning, tightening connections. Renew all fuses if you are unsure of their age. Unplug and re-plug all connections that are easily accessible several times. All these things help.