Most experts seem to recommend painting plastic with a special Krylon Fusion paint but that's not the aged look that you wanted.
A search showed that there are have been questions asked about how to remove coffee stains from white plastic. Depending on the type of plastic, coffee stains might be relatively permanent and there are still ways to remove them even if they're really stubborn. Tea would be another possibility.
Trying to stain the grips by steeping them in strong coffee sounds risky and it may or may not wear off over time.
Anything that doesn't stay on the grips may come off on your hands.
I'm sure that you realize that dyes are usually added to the plastic when it's first molded so coffee staining them would be experimental.
Barkeeper's Friend is one way to remove stains from plastic.
Hint: Since using harsh scrubbers on plastic may cause dirt and stains to accumulate, that knowledge might be used to your advantage if seriously trying to stain.
Another idea would be to use heat but that could promote cracking and what's unknown is how much heat should be used. Would certain hot light bulbs in very close proximity produce enough heat to cause some browning? Will using a propane torch produce any of the desired effect or will it only damage the grips? The effect wouldn't be even and should only be tried in addition to staining unless you're willing to potentially ruin the grips.
Someone said that tumeric produced a bright yellow stain on their white plastic food processor blades. If mixed with brown coffee liquid who knows how it would turn out? But the tumeric yellow probably wouldn't be permanent and may not even take at all.
http://www.ehow.com/way_5179908_coffee-stain-removal-plastic.html
Additional Tips and Ideas
Avoid any harsh scrubbers or cleaners as these can scratch the surface. These scratches will then collect stains and dirt faster making the plastic surface look worse.
The longer stains set on the plastic, the more permanent they will become. Treat stains promptly for the easiest and most successful removal.
http://www.howtocleanstuff.net/house/surfaces/how-to-remove-coffee-stains-from-plastic/