Solozzo was a drugs wholesaler trying to establish a retail network in the US. Prostitution could be a logical starting point for a street-level drugs operation. He also had nightclubs--another prime drugs venue. All of that would be credible to Vito when Sol said the Tatts vouched for him.

I'm guessing that Barzini didn't tell Tatt that he was the real guy behind Sol, at least initially. But once the war broke out, Barzini came out into the open supporting Tatt. Puzo writes (in the Commission meeting scene) that Tattaglia had lost face because the other Dons "knew that his strength had come first from Solozzo and then from the Barzini Family."