Chrome for iOS keeps loggin out of reddit.com
Try logging in to reddit with HTTPS at pay.reddit.com

I can't say for sure if this will work, but at least it's something to try.

pay.reddit.com is HTTPS, not HTTP. Explanation: http://www.reddit.com/r/OutOfTheLoop/comments/1y9543/ive_seen_httpspayredditcomr_a_few_times_what/