Yes definitely, one should shoot with both eyes open. The reasons are many.
One simple resaon is not to change your face's muscular structure which would change your aiming. Else one could use a "pirate's patch" over the other eye, but it is not desirable.
Another more powerful reason is to say that one should try and keep a maximum input of depth and width and volume of light coming from the target and its environment, for perception and information purposes, and all that strenghtens the link between the archer and the target.
But it also happens that one could have an eye-crossover due to fatigue. I noticed that in a friend who was shooting with me this summer. He started to use his glasses and it looks as if it helped him.
A good optometrist would try the following "trick" with glasses : modifying the alignment of one or both lenses in a pair of glasses to modify the eyes' line(s) of sight.
One good thing might be to test for your eye dominance with a fail-safe test to make sure you really are shooting with the correct dominant eye.