The original, Golden Age version of Catwoman (1940-1977) has been repeatedly sent to prison by Batman, escaping each time, and eventually reforming, marrying Bruce Wayne, and bearing their daughter. There was no confinement to Arkham: indeed, Arkham debuted in 1974, well after most of this version of Catwoman's stories had been told. I can find no indication that Earth-Two, the home dimension of these versions of the characters and their stories, had an Arkham.
The Silver Age version of Catwoman (1954-1986) did some jail time; this happened off-stage, and it is unclear that Batman put here there. She was paroled. She was not a patient at Arkham.
The Post-Crisis version of Catwoman (1986-2011) had her origins ret-conned during her run; some versions have her in juvenile detention, not at Batman's hands. This incarnation appears never to have done time as an adult either in prison or Arkham.
The (possibly) current version of Catwoman as of this writing (2011-) has not been in jail. She has, however, done a brief stint in Arkham as part of a Justice League of America plan to infiltrate The Secret Society of Super-Villains. Batman was not involved.
There are many other versions of Catwoman, whose histories I will not exhaustively research; however, the versions listed above represent the bulk of Catwoman's publication history. Batman has imprisoned her. She has been sent to Arkham, though under false pretenses and without Batman's involvement.
Update: In Batman: Killing Time #2 (2022), we see clear indications that Catwoman has done time in Arkham as a genuine patient:
Given what appear to be the current rules of DC continuity, this may only be true within this comic or any comics which make specific reference to it.