In the movie, The Terminal, if officer Dixon had a valid reason, authority, and intent to stop Victor from going to New York, then why was he even bothered to blackmail him (in the name of his 3 friends) when he could have just denied him?
Yes Dixon had the authority to stop Victor from going to New York but no valid reason to do that. His intent to stop Victor from going to New York was purely because of his vindictive nature that made him to misuse his power and not signing on Victor's visa for New York.
After an incidence where out of the goodness of his heart, Victor (by playing a little and innocent trick of translation) helps a desperate Russian man with undocumented drugs for his sick father, Dixon gets pissed off at Victor. Realising that Victor is playing with the rules, Dixon loses his temper at that moment, had a falling-out with him and threatens Victor in front of the Airport Ratings committee (who were there to inspect Dixon's work on which relied his promotion). From that point on Dixon got absolutely irritated with Victor's presence and develops vindictive feelings towards him.
Finally when Victor came to get his visa form signed by Dixon, he (Dixon) instead blackmails him (Victor) into returning to his own country or he'd deport his 3 friends if he (Victor) kept on insisting to get his visa form signed.
So, basically Dixon didn't want Victor to get what he (Victor) wants which is going to New York and since, he (Dixon) didn't have a valid reason to stop him, he resorted to blackmail him (Victor) in the name of his friends.