I haven't had any problems with homebrews going stale, but I'm curious about what people do to minimize introducing oxygen in later steps of the brewing process. If you have a closed brewing system, then that's obviously ideal. But is it possible to pump an inert gas into a secondary carboy before racking? Are there other steps or tips for reducing oxygen in the process?
If you have a kegging setup, you can use it to pump CO2 into your secondary. Or just skip your secondary, which is what most experts recommend these days. No chance of O2 if you just leave it in primary. Purge your kegs with CO2 before and after filling them, too.
What Denny said. CO2 is heavier than oxygen, so if you can pump some into your secondary it will push out the O2.
That said, you may be unnecessarily concerned in some ways. When you transfer to secondary the yeast gets stirred up enough that CO2 will be produced in secondary shortly after racking, and this CO2 will push out any small amounts of O2 that was introduced.
Purging before and after kegging, as Denny already said, is a must though (the keg is closed after transfer and the O2 would be trapped).