You probably knew I'd respond to this one!
A few observations -
1) The "Socrates" of the Clouds bears very little similarity to the real Socrates. I think Aristophanes chose Socrates to "represent" modern philosophy (which he appears to have disliked), and so the fictional Socrates ends up spouting a lot of pseudo-philosophical nonsense, drawing on ideas from the Sophists and the pre-Socratics, that the real Socrates would not have had any time for. This may well have contributed to popular misunderstandings of what Socrates was really about (note, particularly, the claim in "Clouds" that Socrates worshipped a god called "Vortex", and compare it with the charge that Socrates introduced "strange new gods" to Athens.)
2) We must remember that Aristophanes was writing to win prizes, and thus it's not clear whether he invented these silly ideas about Socrates or was simply dealing with prejudices that already existed among the general public. Did he set up these misconceptions about Socrates, did they already exist, or was it a mixture of both? It's impossible to know.
3) If there's any truth in the depiction of relations between Socrates and Aristophanes in Plato's "Symposium", then the two men actually seemed to get on very well. This suggests that Socrates did not object too much to "The Clouds" during his own lifetime.
4) The best "evidence" that The Clouds contributed to anti-Socratic sentiment is Plato's "Apology", where Socrates specifically cites The Clouds as a source of misconceptions about himself. But we don't know how much of the "Apology" genuinely goes back to Socrates and how much is Plato's own rhetorical invention.
Overall, I think it's best to see "The Clouds" as a reflection of the sort of misconceptions and prejudices that led to Socrates' condemnation, rather than trying to blame Aristophanes for his death. As a play it's useful to historians, because it sheds light on some of Athens' popular anti-intellectual currents of thought, which manifested themselves in the purge of intellectuals after the Peloponnesian War, and the sense of religious unease that Athens' philosophers and intellectuals provoked among the uneducated common folk. Aristophanes' play was perhaps not fair or responsible, but I'm willing to forgive him since it's such a damn funny piece of theatre (perhaps my favourite Aristophanes of them all!)