Indeed, saying ‘Bismillah’ or ‘In the Name of Allah’ is recommended when one starts doing anything. However, it is better for the person who tends to perform Ghusl to utter the word ‘Bismillah’ in his heart.
Sheikh Ibn `Uthaymeen, a prominent Saudi Muslim scholar, (may Allah shower mercy on his soul), issued the following fatwa: The opinion which is deemed to be the most correct is that saying ‘Bismillah’ upon performing both Ghusl and ablution is not obligatory. Rather it is recommended. During the course of performing Ghusl in the bathroom, a person should say ‘In the Name of Allah’ in his heart and should not to utter it aloud. So if you are in a similar situation, you have to follow this rule. It is now clear that one should not utter ‘Bismillah’ loudly when he is making Ghusl in the bathroom. Also, there is no specific Du`a’ that is uttered upon performing Ghusl.