Many scholars may Allaah have mercy upon them are of the view that it is permissible to kill the spider as it is harmful, and there is no authentic evidence, as far as we know, about the prohibition of killing it. As regards the narration reported by Abu Nu'ayim may Allaah have mercy upon him and which was narrated by Muhammad Ibn Ibraaheem At-Taymee, that the spider webbed a net on the cave where the Prophet sallallaahu `alayhi wa sallam ( may Allaah exalt his mention ) and Abu Bakr may Allaah be pleased with him were hiding when Quraysh wanted to kill them, and that the Prophet sallallaahu `alayhi wa sallam ( may Allaah exalt his mention ) prohibited to kill it and said that it is among the soldiers of Allaah, then the chain of narrators of this narration is broken; and Muhammad Ibn Ibraaheem At-Taymee may Allaah have mercy upon him did not meet the Prophet sallallaahu `alayhi wa sallam ( may Allaah exalt his mention ).
Therefore, there is no harm on you in killing the spider if you fear that it might harm you.