Travis does so because of
- Social ineptitude
- He is clueless about how inappropriate it is
- But he knows it's inappropriate and likes the idea of angelic Betsy in that dirty theater
Quotes from the screenplay by Paul Schrader.
About Travis (and Betsy)
Travis lives in his own world and has a unique lifestyle separating him from others. He is isolated, especially from people like Betsy. When observing Betsy from his Taxi, he is described
He is like a lone wolf watching the warm campfires of civilization from a distance.
He was in the army unti May 1971 (he may have arrived in NY not too long ago), he is not educated (at the beginning: "Some. Here and there.") and his growing obsession with cleaning the city doesn't help either.
He is unsusual in many ways, e. g. he can't sleep at night, his willingness to "work anywhere, anytime" regardless of danger, just watching TV or later bringing himself back to shape to assassinate Palantine. That's not always bad, though, since it makes Betsy become interested in him in the first place (what a contrast he is to her immature colleague). See scenes 26 and 28, for example, where it's described what draws Betsy to him.
His social ineptitude is also shown in scene 11, when he awkwardly tries to hit on the concession girl of the porn theater
He is obviously trying to be friendly - no easy task for him. God knows he needs a friend.
But overall, he doesn't keep up to date, shows no interest in politics (read scene 27 when Betsy interviews him because he wants to volunteer and he is clueless about Palantine, or scenes 38-39 with Palantine in his cab: "I don't know, sir. I don't follow political issues much.") or culture (not just the porn theater, but also right before, when he gives Betsy the Kris Kristofferson record: "I don't follow music much. I'd like to though. (second thought) Honest."). And he doesn't really care about it, his interest is only due to his interest in Betsy, saying he wants but never does.
In the porn theater
When Betsy realizes that he really wants to take her to a porn movie, she is horrified, but he tells her that many couples go there, so he believes it's socially acceptable.
Travis seems confused. He is so much part of his own world, he fails to comprehend another's world. Compared to the movies he sees, this is respectable. But then there's also something that Travis could not even acknowledge, much less admit: That he really wants to get this pure white girl into that dark porno theatre.
Note the sarcasm in the following:
Travis escorts Betsy to an empty center row. Travis was right. Couples do go to films like this. There are at least six or seven other MEN with their bewigged "DATES".
And when she leaves, the following dialogue
These are not the kind of movies I
Well, I don't follow movies too
You mean these are the only kind of
movies you go to?
The TICKET GIRL watches expressionlessly from the booth.
This is sort of high class...
I mean porno movies.
Note how he admits to not following movies too much. He only knows porn movies.