The tradition of looking down during prayer has been debated within Judaism as well. There is a difference of opinion here. Rabbi Yose stated that eyes should not look up whilst in prayer and that they should look downwards with the head thinking of heaven. (source: Mysticism, Magic and Kabbalah)
But in Islam, the Prophet Muhammed made the decision always to look down during prayer. Believers were not to raise their eyes towards the heavens or sky during prayer. Later the early Jewish position of "prostration" would be restored for prayer.
From the Sunnah of Abu David Solomon, volume 1:
912. Jãbir bin Samurah said:
"Once the Messenger of Allah entered the Masjid and saw people praying with their hands pointing upwards to the skies. He said: 'People should stop raising their eyes to the skies" - Musad-dad (one of the narrators) said: "during prayer" — "or else their eyesight will not return to them." (Sahih)
913. Arias bin Malik narrated that the Messenger of Allah said: "Why do people raise their eyes in the prayers." Then he became even more strict in this regard and said, "They should stop doing so, or else their eyesight will be snatched away from them." (Sahih)
Islam borrowed other customs of the Jews.
From the Jewish Encyclopedia:
But Islam, even more than Christianity, was influenced by the Jewish Forms of Adoration. At first Mohammed commanded that the faces of the faithful should, during prayer, be turned toward Jerusalem; and he only recalled this ordinance when he found that Jews were not to be captured by any such device. The very complicated postures adopted by Moslems at prayer (see Lane's pictures in "Modern Egyptians," i. 75) are probably borrowed from the Jews of Arabia, who, being far removed from Jewish lore, have preserved many archaic customs. These illustrations show all the Forms of Adoration above described as being existent among Jews, and especially that unusual form of sitting solemnly with the head upon the knees.