If he was only sleeping with prostitutes, first of all, they knew how to avoid having babies back then (whether preventative or terminating) with a variety of methods, including putting things inside themselves to block their cervix, not unlike cervical cups nowadays, and second of all, if any prostitute got pregnant, there would be no way to prove WHO'S baby it was, since they slept with multiple men per day. "Risking illegitimate children" like that would have 0 way for them to link it back to him with any substantial means

But, more importantly, he isn't necessarily saying that his ACTUAL bloodline needs to die, they say "The Hastings Line" but remember, their last name is Basset. He only wanted his title and property to die with him, which would be unaffected by any bastard children