I found a box of change in my basement, tucked into a corner on a shelf I didn’t even know was there.
The box is about the size of a shoebox, and there’s no lid. The coins are filthy. Really really filthy. Some have lumps of corroded stuff on them. They seem to be mostly from the 70s and 80s.
I’d like to roll 'em up, but even touching a few leaves me with black fingertips.
How can I wash them to a point where I can either roll them or put them through a coin counting machine?
Assuming none have any collector value (which given your OP sounds correct), then a bucket of warm soapy water, sluice them around a bit, let 'em soak, sluice around, drain, dry.
Would the soapy water get rid of the cancerous looking lumps of dirt + corrosion?
I do a science project with my preschoolers where we wash pennies in a solution of water, vinegar and salt. You could mix up a batch in a bucket and swish it all around. The vinegar and salt combo will ‘scrub’ them clean.
Our mix is about a quarter-cup of water, a tablespoon of vinegar and about a teaspoon of regular table salt. You’ll have to adjust as necessary.
I’d use rubber gloves, too, if they’re that dirty. Yuck.
That’s per kid, for about five pennies.
I’ve laundered money in a similar way as DrDeth recommends, only it was scrubbing it with my bare hands under running water and some dishwashing liquid. Since you mentioned you don’t want your hands dirty you might have to takea more hands-off approach to agitation. But it took care of my sugar-and-syrup encrusted coins with more than a years dirt on them and they looked like brand new.
I’m not sure if that will take care of corrosion but it will probably help immensely with the grime: I was surprised how well it worked. I dont think metal has as many tiny crevices for dirt to hold onto. And if the coins are so corroded that a gentle soap harms them I suspect they would crumble anyway 
You might want to check for dimes and quarters from 1964 and earlier. These have a high silver content (something like 90%) and are worth a lot more than their face value.
Really? Cool. Do you have a cite? What do I do with them when i find one - sell it to a coin dealer or melt it down or what?
Oh and jsgoddess, I have always heard that a baking soda paste (BS and water) was good for cleaning coins. The base chemical might remove some of the corrosion.
Are you going to visit San Francisco anytime soon? The Westin St Francis Hotel has been offering coin cleaning to its guests since 1938.
*Older United States silver coins (1964 and before) are now in demand because of their silver content. (Many were melted down years ago.) At one time these were referred to as - junk silver - because they had no special collector value at the time. Junk silver is still used today to refer to old silver coins. However, times have changed. Today “junk silver coins” are becoming harder to find.
A $1.00 worth (face value) of silver dimes weighs the same (and has the same silver content) as a $1.00 in silver quarters, which also weighs the same as $1.00 in silver half dollars. For example 10 dimes, 4 quarters, or 2 halves all have the same weight and are 90% pure silver composition. A typical dollar face value in US silver coins will contain about 71 % of a troy ounce of pure refined silver. These circulated silver coins are considered by many as a good way to invest in silver. These are sold by some coin dealers in bags of $1,000.00 face value per bag. *
Any coin dealer will buy them.
I can just see me trundling up to the front desk with my shoebox full o’dirt. 
I’ve sold some on eBay and received good prices for them - at least as much as a dealer would have paid. (You’d need at least a decent handful to make this worthwhile.)
What’s fun is if you do this with a bunch of corroded pennies and one nickel. The nickel will get slightly copper-plated.