Write an e-mail to the office asking employees to clean their coffee mugs rather than leave them in the sink. It should be polite, but express some annoyance.
Hello fellow coffee-drinkers,

I love drinking coffee like you, but, I also don't like seeing a bunch of dirty mugs piling up in the sink. We do not have cleaning staff to clean them up during the day. Your coworkers have been cleaning up after you. Please rinse any used coffee mugs and put them in the dishwasher like you do at home! Thank you!