All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
There is nothing wrong with having the name of Baasit. That is because it is permissible to have one of the names that are common between Allaah and His creation. This is proved by the approval of the Prophet, sallallaahu ‘alayhi wa sallam, of some of the names of the Companions that are used for Allaah such as ‘Ali and Hakeem. Also, Baasit can be added to them.
It was stated in Ad-Durr Al-Mukhtaar by Al-Haskafi: "It is permissible to have the names 'Ali and Rasheed, which are common (between Allaah The Almighty and people). That is because their meanings concerning us (human beings) are different from their meanings concerning Allaah The Almighty." [End quote]
For more benefit, please refer to Fataawa 88517 and 94453.
Allaah Knows best.