Tai Chi Chuan has an immediate carryover value to Karate. Swimming is, always, good; and some forms of springboard diving, or conventional gymnastics will help to keep your total body reflexes sharp. Be careful with wrist and hand exercises; it’s very easy to overdo it and inflame tendons or ligaments.
There’s a whole world of: stretching, warm-up, and kata exercises associated with each of the martial arts. This information is available everywhere, today, and should be easy to locate. If I had to chose, just, one form of preparatory/ancillary exercise it would be the, ‘long forms’ of Tai Chi.