Should I worry about galvanic corrosion if using an aluminium shaft inside a stainless steel bearing in something that's going to be washed down frequently?
Considering I will press the shaft into the bearing, is there any way to guard against this? If pressed together in a dry environment, and then I add a bead of silicone sealant at the mating surface (see diagram), will this protect it? 
Thanks
My bigger concern would be the bearing surviving the frequent washdown lol.
A bead of silicone like you mentioned will help with galvanic corrosion, but as others have mentioned will not fully eliminate the issue. The silicone would help by increasing the distance between the two conductive surfaces and reducing the duration and frequency a conductive water solution would complete the circuit touching both the aluminum and stainless steel at the same time. Galvanic corrosion only takes place if there is a completed circuit involving an electrolyte.
One method that may work in this case would be to send off the shaft to have it professionally anodized. This creates an aluminum oxide layer that is not electrically conductive. Then provided this coating is not scratched away when pressed onto the bearing (might need some heat on the bearing or cold on the shaft to make that happen), it should electrically insulate it from the bearing while providing the same mechanical properties. You would then want to add the silicone like you mentioned because there are still pin holes in this coating and is only truly non conductive when dry in that protected area under the bearing.
Other mitigation options would be paint, oil, shields to deflect direct spray, washing with water that has less TDS like reverse osmosis water, or you treat the aluminum as sacrificial and find a way to easily replace it.
