I'd say yes for a couple of reasons, galvanising is a sacrificial protection method if you protect the zinc you will increase the life of the chassis.
Make sure you properly acid etch it before painting it or all the paint will flake off painted zinc lasts very well because it wont corrode like steel and start to lift the paint if it gets chipped.
If you paint it, it doesnt look like a galvanised chassis therefore reducing attention from thiefs!