What is the best and least messy way to clean venetian blinds?

Hey Sharon -- if you blinds are the narrow plastic mini blinds, take down and hang outdoors ( if possible ), close slats, spray with an all purpose cleaner ( like Spic n' Span ); use a soft bristle brush to wipe the slats - sideways - not 'up/down' ... and do both sides; reverse direction of the slats -- close but in the 'up' position; repeat; Rinse with clear water -- preferably from your garden hose to get enough force to really 'rinse' .. open slats all the way ... let hang to dry.

If Plantation/wood slats, there are brushes that will clean each slat individually. -- there are LOTS of different styles available on Amazon,

-- with the mini blinds -- if you can't hang outdoors, I have used a 5 gallon plastic bucket and keeping the blinds drawn up and closed position ... repeatedly 'dunk' in soapy water, reverse ends, repeat and keep changing water .... the rinse thoroughly -- and you can re-hang -- put bath towel on the window sill until they quit dripping .... Sorry -- but this is not 'Not Messy' ... but it works ---

If outside is problematic, you can use the bath tub or shower.

As long as they aren't wood or painted...Close them just enough, so the water can circulate around the slats. Take them off the valance and place them in the whole length of your tub. Add hot water and detergent. Let soak. Change water and repeat if they are stained from smoking or cooking. Then use the shower head to rinse them clean. Drain tub and let them drip dry. Hang them back up with a towel or cloth on the floor if there is still a bit too much water.

You can put an old towel in the bottom of the tub if you are afraid of scratching it.

I've used socks on my hands and wipe each slat - I can't easily take mine down. This works to remove dust, it's very time consuming when they need to be cleaned!