I'm lucky. I have a rinse sink in my laundry room, and I do my mixing and dyeing on the washer/dryer. I have a collection of old t-shirt rags I use to mop up spills, and I keep some terry cloth towels handy to dry my tools on. For direct application dyeing, I generally use some paper towels under the item I'm dyeing, replacing them as I flip the item to dye the other side. Sometimes I use a double rack I made from closet racks (plastic coated with a lip on one side, two wired together to make it like a box). I can dye an item with lots of resist areas with both hands freed up to manipulate the item I'm dyeing. Here is a picture of the rack:

It's also good for allowing pre-soda soaked items to air dry a little before I dye them. I put the lid from a plastic storage bin underneath to catch drips, then I can just rinse and wipe that afterward.