In Hilchot Shabbat 20:14, Rambam implies that asking a non Jew to do a Melacha is a biblical prohibition. However in 6:1 he states it is a rabbinic prohibition? What is the Halacha regarding current day christians and muslims? Is it a rabbinic or biblical prohibition regarding asking them to do a Melacha on Shabbat for a Jew’s benefit?

Asking a non Jew to perform a Melacha on behalf of a Jew is a rabbinic prohibition as you quoted from Rambam Shabbat 6; 1. There is no relevancy to the religion or the belief of the non Jew and the above Halacha applies equally to both Christians and Moslems. In 20; 14 the Rambam talks about a slave or a servant of a Jew that are owned by the Jewish master and have converted partially into Judaism, that is included in the verse of Torah he states there. This is not a practical Halacha nowadays since we have no slaves.