GradSM, I saw a fight choreographer that I work with a lot today (and who has done many, many blood effects) and asked him this question. He said that there is a kind of blood designed to go in eyes, called Kryolan, but it is expensive and more like eye drops (it can create an effect of someone crying blood). His suggestion was to create something with light corn syrup and jello or kool-aid powder, thinned if needed. It really depends on what the blood needs to do- spray, drip, etc.
My current show uses Reel blood, original version, thinned with water and it sprays in actor's face without bothering his eyes, but it is not designed to get in your eyes- the bottle instructions says to flush eyes if you get any blood in them. It is a little sticky but not hard to wash off of scenery, and comes out of clothes with soap and a little OxyClean.