In this context, what does "halfway respectable" mean?
He was smooth and he made it all seem halfway respectable, but it came down to prostitution and we both knew it.
Source: Single White Female by John Lutz
This is a common English expression indicating that something is a bit (or perhaps a lot) shady. It means someone is trying to make something that is illegal or immoral appear as if it is reasonably acceptable. For example, calling a brothel a gentleman's club; or calling a Ponzi scheme a 'once-in-a-lifetime investment opportunity'; or calling a bribe a business facilitation fee.
A similar English expression is 'halfway decent', although this is almost always used in a more positive way. When used regarding food, (eg 'That cafe makes a halfway decent cup of coffee'), this is a complement meaning that the coffee is pretty good. If you are about to enter someone's bedroom (eg a parent entering a child's room) you may ask, 'Are you halfway decent?" when you knock on the door. The meaning here is 'Are you sufficiently dressed that I won't embarrass you (or me) when I enter your room'.
"Halfway respectable" means exactly what it says: somewhere between completely respectable and totally disreputable. It implies an effort has been made to make matters look good, but that the effort has not been completely successful. In the sentence quoted by the OP the "he" has woven a tale that makes it sound OK for someone to take money for sex in certain circumstances, but the tale has not deceived the narrator.
In the door-knocking example, the person knocking might ask "are you decent" and get the reply "I am halfway decent" implying that they are permitted to enter.