In which military branch did Ted Striker serve?

Its inconsistently pictured but when mentioned it is the Air Force

His voiceover describes him as being in the Air Force

        STRIKER (v.o.)
I was in the Air Force, stationed in
Drambuie, on the Barbary Coast. I used to
hang out in the Magumba Bar.

Later when asked to fly the plane:

        STRIKER
Well, I flew single-engine fighters in the
Air Force, but this plane has four
engines. It's an entirely different kind
of flying...all together!!!

And when reporters ask who is flying the plane:

        REPORTER #2
Who's flying the plane?


        McCROSKEY
One of the passengers. But he's an
experienced air force pilot who flew
during the war so there's no cause for
alarm.

Yet he is wearing a Navy dress uniform in the Saturday Night Fever scene, and in an Army Mental Health Hospital in yet another.

This is almost certainly artistic license. He shouldn't be wearing a dress uniform in a bar anyway - its just there so he can have white pants on and look the part for dancing during Saturday Night Fever song and the Air Force dress uniform is blue. For the Hospital scene they want him painting the baby rescue scene, so set it in the Army.

Its only important to the plot that he is an ex-military pilot with PTSD. It's not super important to the writers that this is in any way consistent. The plane dogfight clips used appear to show World War II or Korean War era planes, which he is far too young to have fought in. The writers and directors are just turning the character into a stereotype for the joke potential.

While the accepted answer is indeed correct in the context of Airplane!, please note that the film wasn’t just a parody of the Airport action/drama films of the 1970s, but also a direct parody of two earlier films: Flight into Danger (1956) and Zero Hour! (1957).

In the case of Zero Hour! the character of Ted Striker (Airplane!) is directly based on the character of Ted Stryker Zero Hour!.

And as per the Wikipedia description of the plot of that film:

“During the closing days of the Second World War, six members of the Royal Canadian Air Force fighter squadron led by pilot Ted Stryker (Dana Andrews) are killed because of a command decision made by him. Years later, in civilian life in Winnipeg, Manitoba, a guilt-stricken Stryker goes through many jobs, and his marriage is in trouble.”

While PTSD is not explicitly stated, the description of Stryker’s World War II experience clearly implies that. And Stryker is clearly described as a member of the Royal Canadian Air Force here as well.

Yes, Airplane! parody of the Airport film series, but it was also a direct parody of the plots of Zero Hour! and Flight into Danger. As such, the role/character of “Ted Striker” is clearly based on the role/character of “Ted Stryker.” And Ted Striker is an Air Force pilot.