You do not have to upgrade them to 64bit OS - QueueManager runs on 32bit OS.
Neither does the initiator have to run on 64bit OS, though as you have one it's the sensible thing to do.

(edited to correct error)

"If I were a shadow, I know I wouldn't like to be half of what I should be."
Mr Otsuka, the old black tomcat in Kafka on the Shore (Haruki Murakami)