The film is (probably) set in the 23rd century.
The slightly altered David Bowie lyrics for the song "I've not been to Paradise" used in the high school dance towards the end of the film contains the line:
"All is well. Watch the days go by. All is well. 23rd century dies."
And according to Wikipedia, the director has confirmed this explicitly:
Starship Troopers is placed in the 23rd century, as said by Paul Verhoeven in the director's commentary.
The only commentary I've found on Youtube is this one by Verhoeven and screenwriter Edward Neumeier, but Verhoeven also did at least one other commentary together with some of the acting cast, as referenced here. I haven't found a video or transcript of this commentary though to confirm the 23rd-century reference.