When training snow boarders, what are the most effective exercises for balance and core strength?

First everthing must be done standing, it is a standing sport. This alone will challenge the core. Make sure that the inntrinsic core is firing with the extrinsic core and there are no compensation patterns. If all that is fine then the next level is doing everything in the workout on a single leg then progress that through different modalities of progression. Such as, Airex pad, half foam rolloer, dyna disk, bosu, and voodoo board. This will train stabilty integrated with movement patterns. If extra core work is required then go with planks all around, some Gray Cook chop and lift patterns, and good ol deadlifts for posterior chain strength.