When I am doing hand work that does not require concentration I pray and talk to God about what is on my heart.
I cannot think of a Bible verse to directly support the closing of our eyes in prayer.
But, when Jesus was in the garden of Gethsemane, He "fell on his face, and prayed" (Matt 26:39). In that situation I would think that His eyes were closed.
(Ellen White speaks directly on the subject. I can post that thought if you'd like to see it Buluba.)
The Bible is silent on this question - even when Jesus directly said to His disciples "this is the way you should pray" (the Lord's Prayer), He did not mention eyes open or closed, or position of the body.
Shall we become like the Muslims or Catholics, who must sit or stand or kneel or make certain gestures in a particular order in order for the prayer to be valid? Shall we develop strict rules and procedures for washing before or after prayer, what clothing should be worn, what places we may or must not pray? Shall we exclude people with certain physical infirmities (like discharges, in the Old Testament, or menstruating women, like many religions to this day)?
I agree with other poster here that it depends on the circumstances.
I pray to God in all circumstances. Most of the time it is with eyes wide open. We are to pray without ceasing. So, it would be hard to stumble around through your day with your eyes closed. :)