Surprisingly, efficient dishwashers often use less water than washing by hand. According to Green Choices, dishwashers use three to four times less water than hand washing when cleaning a twelve-place table setting.
In terms of carbon footprint, The Guardian quotes these figures by Mike Berners-Lee:
'Almost zero CO2e: by hand in cold water (but the plates aren't clean)
540g CO2e: by hand, using water sparingly and not too hot
770g CO2e: in a dishwasher at 55°C
990g CO2e: in a dishwasher at 65°C
8000g CO2e: by hand, with extravagant use of water’
As these figures show, if dishes are to be properly cleaned, the dishwasher is probably the safer option. If hand washing is the only option, then try to create a lather of washing up liquid before rinsing items sparingly, instead of filling up a bowl with hot water that goes dirty and has to be refilled.
If you do have a dishwasher, make sure to run it on an eco setting (if it has one), scrape - but don't rinse - plates beforehand, only put it on when it is completely full, and try timing it to run at night when energy demand is at its lowest.
Berners-Lee, Mike, and Duncan Clark, 'What's the carbon footprint of... doing the dishes?', The Guardian (August 2010),<https://www.theguardian.com/environment/green-living-blog/2010/aug/19/carbon-footprints-dishwasher-washing-up>
Green Choices, 'Is it greener to hand wash or use a dishwasher?' (2020), <https://www.greenchoices.org/green-living/at-home/is-it-greener-to-hand-wash-or-use-a-dishwasher>